The pooled variance is the weighted average of the variances of two or more groups, where the weights are the degrees of freedom (n-1) for each group.
To get the pooled variance for the given samples, we need to find the variance of each sample and plug in the values in the formula above. Sample 1 has n = 8
and ss = 168.
To get the variance of this sample (S1²), Plugging in the values Now let's find the variance of sample 2. It has n = 6 and ss = 120.
Therefore, the pooled variance for the given two samples is 24. The pooled variance for the given two samples is 24. The pooled variance is the weighted average of the variances of two or more groups, where the weights are the degrees of freedom (n-1) for each group. We can find the variance of each sample using the formula S² = SS/(n-1), where SS is the sum of squares and n is the sample size. Plugging in the values, we find that the variance of both samples is 24. Finally, we can use the formula Sp² = (S1²(n1-1) + S2²(n2-1))/(n1+n2-2) to find the pooled variance, which is also 24.
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A centripetal force of 180 n acts on a 1,450-kg satellite moving with a speed of 4,500 m/s in a circular orbit around a planet. what is the radius of its orbit?
The radius of the satellite's orbit is approximately 163,402,777.8 meters.
The centripetal force acting on the satellite is 180 N. We know that the centripetal force is given by the formula Fc = (mv^2)/r, where Fc is the centripetal force, m is the mass of the satellite, v is the velocity, and r is the radius of the orbit.
In this case, we are given the mass of the satellite as 1,450 kg and the velocity as 4,500 m/s. We can rearrange the formula to solve for r:
r = (mv^2) / Fc
Substituting the given values, we have:
r = (1450 kg * (4500 m/s)^2) / 180 N
Simplifying the expression:
r = (1450 kg * 20250000 m^2/s^2) / 180 N
r = (29412500000 kg * m^2/s^2) / 180 N
r ≈ 163402777.8 kg * m^2/Ns^2
The radius of the satellite's orbit is approximately 163,402,777.8 meters.
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How long it takes for the light of a star to reach us if the
star is at a distance of 8 × 10^10km from Earth.
It takes approximately 2.67 × 10⁸ seconds for the light of a star to reach us from a distance of 8 × 10¹⁰ km.
The time it takes for the light of a star to reach us can be calculated using the formula t = d/c, where t is the time, d is the distance, and c is the speed of light.
In this case, the star is at a distance of 8 × 10¹⁰ km from Earth. To convert this distance to meters, we multiply by 10^6 since 1 km is equal to 10³ meters. So the distance in meters is 8 × 10¹⁶ meters.
The speed of light (c) is approximately 3 × 10⁸ meters per second. Plugging these values into the formula, we get
t = (8 × 10¹⁶ meters) / (3 × 10⁸ meters per second). Simplifying this expression gives us t ≈ 2.67 × 10⁸ seconds.
Therefore, it takes approximately 2.67 × 10⁸ seconds for the light of a star to reach us from a distance of 8 × 10¹⁰ km.
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An infrared thermometer (or pyrometer) detects radiation emitted from surfaces to measure temperature. Using an infrared thermometer, a scientist measures a person's skin temperature as 32.7°C.What is the wavelength (in µm) of photons emitted with the greatest intensity from the person's skin? (Enter your answer to at least two decimal places.)
The wavelength (in µm) of photons emitted with the greatest intensity from the person's skin is 9.47 µm
The peak wavelength of the photons emitted by an object is calculated using Wien's displacement law.
Infrared thermometers detect radiation from surfaces and measure temperature.
Using an infrared thermometer, a scientist measures a person's skin temperature as 32.7°C.
We're being asked to figure out the wavelength (in µm) of photons emitted with the greatest intensity from the person's skin.
We can use Wien's displacement law to find the wavelength that corresponds to the maximum intensity of the radiation emitted by the person's skin.
The equation is given by:
λmax = b/T
where b = 2.898 × 10^-3 m K is Wien's displacement constant, and T is the absolute temperature of the object.
We must first convert the skin temperature from degrees Celsius to Kelvin.
Temperature in Kelvin (K) = Temperature in Celsius (°C) + 273.15K
= 32.7°C + 273.15K
= 305.85K
λmax = b/T
= (2.898 × 10^-3 m K)/(305.85 K)
= 9.47 × 10^-6 m
= 9.47 µm
Therefore, the wavelength (in µm) of photons emitted with the greatest intensity from the person's skin is 9.47 µm.
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The ims current in an RL cecut is 032 A when it is connected to an ac generator with a frequency of 60 He and an rms voltage of 40 V Part A Given that the inductor has an inductance of 120 ml, what is the resistance of the resistor? Express your answer using two significant figures. VAX ? R Submit Request Answer Part B Find the rms voltage across the resistor Express your answer using two significant figures. VE ΑΣΦ Submit Best An ? D V Submit Request Answer Part C Find the rms voltage across the inductor. Express your answer using two significant figures. 15] ΑΣΦ → www. Vrms,L= Submit Request Answer Part D Complete previous part(s) ▼ ? V
Part A) The resistance of the resistor is approximately 125 Ω, Part B) The rms voltage across the resistor is approximately 40 V, Part C) The rms voltage across the inductor is approximately 45.24 V and Part D) The rms voltage across the resistor and inductor, which are 40 V and 45.24 V, respectively.
Part A:
To find the resistance of the resistor in the RL circuit, we can use Ohm's law:
V = I * R
Where V is the voltage, I is the current, and R is the resistance.
Given that the current I = 0.32 A and the voltage V = 40 V, we can rearrange the equation to solve for R:
R = V / I
R = 40 V / 0.32 A
R ≈ 125 Ω
Therefore, the resistance of the resistor is approximately 125 Ω.
Part B:
The voltage across the resistor in an RL circuit can be determined by multiplying the current and the resistance:
Vr = I * R
Vr = 0.32 A * 125 Ω
Vr ≈ 40 V
Therefore, the rms voltage across the resistor is approximately 40 V.
Part C:
To find the rms voltage across the inductor, we can use the relationship between voltage, current, and inductance in an RL circuit:
Vl = I * XL
Where Vl is the voltage across the inductor and XL is the inductive reactance.
The inductive reactance XL can be calculated using the formula:
XL = 2πfL
Where f is the frequency and L is the inductance.
Given that the frequency f = 60 Hz and the inductance L = 120 mH (or 0.12 H), we can calculate XL:
XL = 2π * 60 Hz * 0.12 H
XL ≈ 45.24 Ω
Therefore, the rms voltage across the inductor is approximately 45.24 V.
Part D:
The previous parts have already provided the answers for the rms voltage across the resistor and inductor, which are 40 V and 45.24 V, respectively.
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Your new weed-cutter requires, as fuel, a gas-to-oil mixture of 23-to-1 (23 parts of gas mixed with one part of oil). You have 2.2 gallons of gas. How much oil, in gallons, should you add
To achieve the gas-to-oil mixture of 23-to-1 with 2.2 gallons of gas, you should add approximately 0.0957 gallons of oil.
To determine how much oil should be added to the 2.2 gallons of gas for the gas-to-oil mixture of 23-to-1, we need to calculate the ratio of gas to oil.
The ratio of gas to oil is given as 23-to-1, which means for every 23 parts of gas, 1 part of oil is required.
Let's calculate the amount of oil needed:
Oil = Gas / Ratio
Oil = 2.2 gallons / 23
Oil ≈ 0.0957 gallons
Therefore, you should add approximately 0.0957 gallons of oil to the 2.2 gallons of gas to achieve the gas-to-oil mixture of 23-to-1.
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A smoke particle with a mass of 25 ug and charged at -9.0x10-1* C is falling straight downward at 2.0 mm/s, when it enters a magnetic field of 0.50 T pointed directly South. Determine the magnetic force (magnitude and direction) on the particle.
The magnitude of the magnetic force on the smoke particle is 9.0x10^(-4) N with the direction of the force towards the East.
To determine the magnetic force on the smoke particle, we can use the equation F = qvB, where F is the force, q is the charge of the particle, v is its velocity, and B is the magnetic field strength.
Given that the charge of the smoke particle is -9.0x10^(-1) C, its velocity is 2.0 mm/s (which can be converted to 2.0x10^(-3) m/s), and the magnetic field strength is 0.50 T, we can calculate the magnetic force.
Using the equation F = qvB, we can substitute the values: F = (-9.0x10^(-1) C) x (2.0x10^(-3) m/s) x (0.50 T). Simplifying this expression, we find that the magnitude of the magnetic force on the particle is 9.0x10^(-4) N.
The direction of the magnetic force can be determined using the right-hand rule. Since the magnetic field points directly South and the velocity of the particle is downward, the force will be perpendicular to both the velocity and the magnetic field, and it will be directed towards the East.
Therefore, the magnitude of the magnetic force on the smoke particle is 9.0x10^(-4) N, and the direction of the force is towards the East.
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A beam of green light enters glass from air, at an angle of incidence = 39 degrees. The frequency of green light = 560 x 1012 Hz. Refractive index of glass = 1.5. Speed of light in air = 3 x 108 m/s. What will be its wavelength inside the glass? Write your answer in terms of nanometers. You Answered 357 Correct Answer 804 margin of error +/- 3%
The wavelength of green light inside the glass is approximately 357 nanometers, calculated using the given angle of incidence, frequency, and refractive index. The speed of light in the glass is determined based on the speed of light in air and the refractive index of the glass.
To find the wavelength of light inside the glass, we can use the formula:
wavelength = (speed of light in vacuum) / (frequency)
Given:
Angle of incidence = 39 degrees
Frequency of green light = 560 x 10¹² Hz
Refractive index of glass (n) = 1.5
Speed of light in air = 3 x 10⁸ m/s
First, we need to find the angle of refraction using Snell's Law:
n₁ * sin(angle of incidence) = n₂ * sin(angle of refraction)
In this case, n₁ is the refractive index of air (approximately 1) and n₂ is the refractive index of glass (1.5).
1 * sin(39°) = 1.5 * sin(angle of refraction)
sin(angle of refraction) = (1 * sin(39°)) / 1.5
sin(angle of refraction) = 0.5147
angle of refraction ≈ arcsin(0.5147) ≈ 31.56°
Now, we can calculate the speed of light in the glass using the refractive index:
Speed of light in glass = (speed of light in air) / refractive index of glass
Speed of light in glass = (3 x 10⁸ m/s) / 1.5 = 2 x 10⁸ m/s
Finally, we can calculate the wavelength inside the glass using the speed of light in the glass and the frequency of the light:
wavelength = (speed of light in glass) / frequency
wavelength = (2 x 10⁸ m/s) / (560 x 10¹² Hz)
Converting the answer to nanometers:
wavelength ≈ 357 nm
Therefore, the wavelength of the green light inside the glass is approximately 357 nanometers.
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The uniform plane wave in a non-magnetic medium has an electric field component: E-10 cos (2x10 t-2z) a, V/m. The wave propagation constant k and wavelength λ are given by: (a) π,2 (b) 2,π (c) 2X10, π (d) π, 2X108
The intrinsic impedance (in ohms) of the EM wave propagating in a non-magnetic medium with electric field E described in Q16 is given by: (a) 1207, (approx. 377) (b) 40. (approx. 126) (c) 807, (approx. 251)
The equation of electric field is given as: E = E-10 cos (2x10 t-2z) a, V/m. Here, E0 = 10 V/m. The equation of wave propagation constant k and wavelength λ can be given as:k = 2π/λ ...(1)According to the problem,λ/k = λ/2π = 2π/k= v,where v is the velocity of propagation of EM wave in non-magnetic medium.
The equation of intrinsic impedance (η) of the EM wave propagating in a non-magnetic medium is given as:η = √μ0/ε0,where μ0 is the permeability of free space and ε0 is the permittivity of free space. So, the value of intrinsic impedance (η) can be found as:η = √μ0/ε0 = √4π × 10⁻⁷/8.854 × 10⁻¹² = √1.131 × 10¹⁷ = 1.064 × 10⁹ Ω.The option that correctly represents the intrinsic impedance of the EM wave propagating in a non-magnetic medium is (c) 807 (approx. 251).
Thus, the correct option is (c).Note: Intrinsic impedance (η) of a medium is a ratio of electric field to the magnetic field intensity of the medium. In free space, the intrinsic impedance of a medium is given as:η = √μ0/ε0 = √4π × 10⁻⁷/8.854 × 10⁻¹² = 376.7 Ω or approx. 377 Ω.
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Question 18 What is sea level pressure? a 1013.2 Pa b. 1012. 3 mb c. 1032 mb d. 1013.2 mb
Sea level pressure is the pressure that would be measured by a barometer at sea level, and is typically expressed in millibars (mb) or hectopascals (hPa). It varies depending on weather conditions and can range from around 950 mb to 1050 mb (option d).
The pressure is the amount of force exerted per unit area. A force of 1 newton applied over an area of 1 square meter is equivalent to a pressure of 1 pascal (Pa). In meteorology, pressure is usually measured in millibars (mb) or hectopascals (hPa).What is sea level pressure?Sea level pressure is the atmospheric pressure measured at mean sea level.
Sea level pressure is used in weather maps and for general weather reporting. It is a convenient way to compare the pressure at different locations since it removes the effect of altitude on pressure. The correct option is d.
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a girl at a state fair swings a ball in vertical circle at the end of a string. the force of the bottom of the string is?
When a ball is swung on a string in a vertical circle, the tension is greatest at the bottom of the circular path. This is where the rope is most likely to break. It should make sense that the tension at the bottom is the greatest.
What is the energy required to transition from n=1 to n=2 in a Lithium atom with only one electron? Remember, for Lithium, Z=3. eV Submit Answer Tries 0/2 What is the corresponding wavelength of light in nm? nm Submit Answer Tries 0/2 Can you see this EM radiation? IncorrectYes. Correct: No, it is too high of energy to see. IncorrectNo, it is too low of energy to see. Computer's answer now shown above. You are correct. Your receipt no. is 164-4692 ? Previous Tries
The energy required for this transition is approximately 30.6 eV. The corresponding wavelength of the emitted light is approximately 12.86 nm. Ultraviolet light falls within a specific wavelength range that is not visible to the human eye because it is shorter than visible light.
To calculate the energy required for the transition from n=1 to n=2 in a lithium atom with only one electron, we can use the formula for the energy of an electron in a hydrogen-like atom:
E = -13.6 * Z² / n²
Where E is the energy, Z is the atomic number, and n is the principal quantum number.
For lithium (Z=3), the energy for the transition from n=1 to n=2 is:
E = -13.6 * 3² / 2² = -13.6 * 9 / 4 = -30.6 eV
Therefore, the energy required for this transition is approximately 30.6 eV.
To find the corresponding wavelength of light emitted, we can use the energy-wavelength relationship:
E = hc / λ
Where E is the energy, h is Planck's constant (approximately 4.136 x 10⁻¹⁵ eV s), c is the speed of light (approximately 2.998 x 10⁸ m/s), and λ is the wavelength.
Solving for λ:
λ = hc / E = (4.136 x 10⁻¹⁵ eV s * 2.998 x 10⁸ m/s) / 30.6 eV
Calculating this, we find:
λ ≈ 12.86 nm
Therefore, the corresponding wavelength of the emitted light is approximately 12.86 nm.
This wavelength falls within the ultraviolet (UV) region of the electromagnetic spectrum. UV light is not visible to the human eye as its wavelengths are shorter than those of visible light (approximately 400-700 nm). So, we cannot see this specific electromagnetic radiation emitted during the transition from n=1 to n=2 in a lithium atom.
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A resistor and a capacitor are in series with an AC source. The impedance is Z=10.4Ω at 450 Hz and Z=16.6Ω at 180 Hz. Find R and C.
The values of R and C are approximately R = 3.76 Ω and C ≈ 2.18 x 10⁻⁶ F, respectively.
For finding the values of resistance (R) and capacitance (C), using the formulas for the impedance of a resistor (ZR) and a capacitor (ZC) in an AC circuit.
The impedance of a resistor (ZR) is given by ZR = R, where R is the resistance value.
The impedance of a capacitor (ZC) is given by ZC = 1 / (2πfC), where f is the frequency in hertz (Hz) and C is the capacitance value.
Given,
Z = 10.4 Ω at 450 Hz
Z = 16.6 Ω at 180 Hz,
For 450 Hz:
Z = ZR + ZC
10.4 = R + 1 / (2π ×450 × C)
For 180 Hz:
Z = ZR + ZC
16.6 = R + 1 / (2π ×180 × C)
From the first equation:
10.4 = R + 1 / (900πC)
10.4 × (900πC) = R × (900πC) + 1
9360πC² = 900πCR + 1
From the second equation:
16.6 = R + 1 / (360πC)
16.6 ×(360πC) = R × (360πC) + 1
5976πC² = 360πCR + 1
Now, equate the two equations:
9360πC² = 5976πC²
3384πC² = 900πCR
C² = (900/3384)CR
Since C²= CR, substitute this into the equation:
C² = (900/3384)C²R
Divide both sides by C²:
1 = (900/3384)R
R = 3384/900
R = 3.76 Ω
Substituting R = 3.76:
10.4 = 3.76 + 1 / (900πC)
6.64 = 1 / (900πC)
900πC = 1 / 6.64
C = 1 / (6.64 ×900π)
C ≈ 2.18 x 10⁻⁶ F
Therefore, the values of R and C are approximately R = 3.76 Ω and C ≈ 2.18 x 10⁻⁶ F, respectively.
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Your sister weights 725 N on Earth (g=9. 80 m/s^2). If you take her to the Mars (g=3. 72 m/s^2) find her mass on Mars
According to the given statement , your sister's mass on Mars is approximately 74.0 kg.
To find your sister's mass on Mars, we can use the formula:
Weight = Mass * Acceleration due to gravity
First, let's calculate your sister's mass on Earth using the given weight and acceleration due to gravity:
Weight on Earth = 725 N
Acceleration due to gravity on Earth = 9.80 m/s²
Using the formula, we can rearrange it to solve for mass:
Mass on Earth = Weight on Earth / Acceleration due to gravity on Earth
Substituting the values, we get:
Mass on Earth = 725 N / 9.80 m/s²
Calculating this, we find that your sister's mass on Earth is approximately 74.0 kg.
Next, let's calculate your sister's mass on Mars using the given weight and acceleration due to gravity:
Weight on Mars = ?
Acceleration due to gravity on Mars = 3.72 m/s²
Using the same formula, we can rearrange it to solve for mass:
Mass on Mars = Weight on Mars / Acceleration due to gravity on Mars
We know that weight is directly proportional to mass, so the ratio of the weights on Mars and Earth will be the same as the ratio of the masses on Mars and Earth:
Weight on Mars / Weight on Earth = Mass on Mars / Mass on Earth
Substituting the known values, we have:
Weight on Mars / 725 N = Mass on Mars / 74.0 kg
Simplifying this equation, we can cross multiply:
Weight on Mars * 74.0 kg = 725 N * Mass on Mars
Dividing both sides of the equation by 725 N, we get:
Weight on Mars * 74.0 kg / 725 N = Mass on Mars
Finally, substituting the given values, we can calculate your sister's mass on Mars:
Mass on Mars = (725 N * 74.0 kg) / 725 N
Simplifying this, we find that your sister's mass on Mars is approximately 74.0 kg.
Therefore, your sister's mass on Mars is approximately 74.0 kg.
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A model airplane with mass 0.750 kg is tethered to the ground by a wire so that it flies in a horizontal circle 30.0m in radius. The airplane engine provides a net thrust of 0.800N perpendicular to the tethering wire.(b) Find the angular acceleration of the airplane.
The angular acceleration of the airplane is 0.0356 rad/s².
To find the angular acceleration of the airplane, we can use the equation:
Net force = mass × radius × angular acceleration
Given that the net force is 0.800N and the mass of the airplane is 0.750 kg, we can rearrange the equation to solve for angular acceleration.
Angular acceleration = Net force / (mass × radius)
Substituting the given values:
Angular acceleration = 0.800N / (0.750 kg × 30.0m)
Calculating this gives us:
Angular acceleration = 0.800N / 22.5 kg·m/s²
Simplifying further, the angular acceleration is:
Angular acceleration = 0.0356 rad/s²
Therefore, the angular acceleration of the airplane is 0.0356 rad/s². This means that the airplane is accelerating angularly at a rate of 0.0356 radians per second squared..
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a) Consider the motion of a spin particle of mass m in a potential well of length +00 2L described by the potential ro 0 SX S2L VO elsewhere i) For this particle, determine the energy levels and the corresponding eigenfunctions and specify their degeneracy. [6 marks] ii) The spinį particle is now placed in a weak magnetic field Bu. Bu, 03XL B where u, and u are the unit vectors in the x LSX S2L and z directions respectively. This field gives rise to the perturbation À, = -A:B , where û My is the Bohr magneton and Ŝ is the spin A operator. Use time-independent degenerate perturbation theory to calculate the energy shifts due to the magnetic field to first order in the field. 2μες, [17 marks] b) Consider now two non-interacting spin particles in the unperturbed potential well of length 2L. Calculate the lowest possible energy for the triplet spin state and the spatial part of the corresponding wavefunction. [7 marks)
Consider the motion of a spin particle of mass m in a potential well of length +00 2L described by the potential[tex]V(0) = 0, V(x) = ∞, V(±2L) = ∞, V(x) = VO[/tex] elsewhere.
The time-independent Schrödinger's equation for a system is given as:Hψ = EψHere, H is the Hamiltonian operator, E is the total energy of the system and ψ is the wave function of the particle. Hence, the Schrödinger's equation for a spin particle in the potential well is given by[tex]: (−ћ2/2m) ∂2ψ(x)/∂x2 + V(x)ψ(x) = Eψ(x)[/tex]Here.
Planck constant and m is the mass of the particle. The wave function of the particle for the potential well is given as:ψ(x) = A sin(πnx/2L)Here, A is the normalization constant and n is the quantum number. Hence, the energy of the particle is given as: [tex]E(n) = (n2ћ2π2/2mL2) + VO[/tex] (i) For this particle.
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discuss why we introduced the interaction picture for
time-dependent perturbation theory
The interaction picture is introduced in time-dependent perturbation theory to separate the effects of the unperturbed system and the perturbation, simplifying calculations. It allows for easier analysis of time-dependent perturbations by transforming the state vectors and operators according to a unitary transformation.
The interaction picture is introduced in time-dependent perturbation theory to simplify the analysis of systems undergoing time-dependent perturbations. In this picture, the Hamiltonian of the system is split into two parts: the unperturbed Hamiltonian and the perturbation Hamiltonian.
The unperturbed Hamiltonian describes the system's behavior in the absence of perturbation, while the perturbation Hamiltonian accounts for the time-dependent perturbation.
By working in the interaction picture, we can separate the time evolution due to the unperturbed Hamiltonian from the effects of the perturbation. This separation allows us to treat the perturbation as a small correction to the unperturbed system, making the calculations more manageable.
In the interaction picture, the state vectors and operators are transformed according to a unitary transformation to account for the time evolution due to the unperturbed Hamiltonian. This transformation simplifies the time dependence of the operators and allows for easier calculations of expectation values and transition probabilities.
Overall, the introduction of the interaction picture in time-dependent perturbation theory provides a convenient framework for studying the effects of time-dependent perturbations on quantum systems and simplifies the mathematical analysis of the problem.
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. Your derived formula should have a similar form as the empirical Balmar formula: 1 λ = RH 1 2 2 − 1 n2 (2) where n = 3, 4, 5, 6 can be used to calculate the four visible lines of Hydrogen spectrum, and RH = 1.097 × 107m−1 . Identify RH in terms of E0, h, c in the formula you derived and calculate its value (check that you get units of m−1 as well).
The derived formula relates the wavelength of the hydrogen spectrum to the Rydberg constant (RH). By substituting the specific values of E0, h, and c, RH is calculated to be approximately 1.097 × 10^7 m^(-1).
To calculate the value of RH in the derived formula, we need the specific values of E0, h, and c.
The ground state energy of the hydrogen atom (E0) is approximately -13.6 eV or -2.18 × 10^(-18) J.
The Planck's constant (h) is approximately 6.626 × 10^(-34) J·s.
The speed of light (c) is approximately 2.998 × 10^8 m/s.
Now we can substitute these values into the equation:
RH = E0 / (h * c)
= (-2.18 × 10^(-18) J) / (6.626 × 10^(-34) J·s * 2.998 × 10^8 m/s)
Performing the calculation gives us:RH ≈ 1.097 × 10^7 m^(-1)
Therefore, the value of RH in the derived formula is approximately 1.097 × 10^7 m^(-1).
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The position of a body is given by x(t) = t2-4t+9. What is the body's acceleration at t = 0?
The speed of a body is given by v(t) = 2t. How far has the body moved from t = 0 to t = 1?
We drop a rock from a height of 3.0 meters above the ground. At what speed does the stone hit the ground?
We throw a stone straight up, the stone comes 12m up. How long did the stone take up?
The body's acceleration at t = 0, we substitute t = 0 into the expression for acceleration: a(0) = 2. And The distance traveled by the body from t = 0 to t = 1, we need to integrate the speed function over the given time interval. Also, The speed at which the rock hits the ground when dropped from a height of 3.0 meters, is 1.566 seconds to reach a height of 12 m.
To find the body's acceleration at t = 0, we need to differentiate the position function x(t) with respect to time: x(t) = t^2 - 4t + 9
Differentiating x(t) with respect to t, we get:
v(t) = 2t
Differentiating v(t) with respect to t again, we find the acceleration function:
a(t) = 2
Therefore, the body's acceleration at t = 0 is 2.
To find how far the body has moved from t = 0 to t = 1, we need to integrate the speed function v(t) over the interval [0, 1]:
v(t) = 2t
Integrating v(t) with respect to t, we get the displacement function:
s(t) = t^2
To find the distance traveled from t = 0 to t = 1, we evaluate the displacement function at t = 1 and subtract the displacement at t = 0:
s(1) - s(0) = 1^2 - 0^2 = 1 - 0 = 1
Therefore, the body has moved 1 unit of distance from t = 0 to t = 1.
When a rock is dropped from a height of 3.0 meters above the ground, its initial velocity is 0 m/s. Using the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration due to gravity (-9.8 m/s^2), and s is the displacement.
We have:
v = ?
u = 0 m/s
a = -9.8 m/s^2
s = -3.0 m (negative because the displacement is downward)
Plugging in the values, we can solve for the final velocity:
v^2 = (0 m/s)^2 + 2(-9.8 m/s^2)(-3.0 m)
v^2 = 0 + 58.8
v = √58.8 ≈ 7.67 m/s
Therefore, the stone hits the ground with a speed of approximately 7.67 m/s.
To determine the time it takes for the stone to reach a height of 12 m, we can use the equation of motion:
s = ut + (1/2)at^2
where s is the displacement, u is the initial velocity, a is the acceleration due to gravity (-9.8 m/s^2), and t is the time.
We have:
s = 12 m
u = ?
a = -9.8 m/s^2
t = ?
At the highest point, the velocity is 0 m/s, so u = 0 m/s.
Plugging in the values, we can solve for the time:
12 m = 0 m/s * t + (1/2)(-9.8 m/s^2)(t^2)
12 m = -4.9 m/s^2 * t^2
t^2 = -12 m / -4.9 m/s^2
t^2 ≈ 2.449 s^2
t ≈ √2.449 ≈ 1.566 s
Therefore, the stone takes approximately 1.566 seconds to reach a height of 12 m.
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A particle moves through an xyz coordinate system while a force acts on it. When the particle has the position vector 7 = (2.00 mi - (3.00 m)ſ + (2.00 m), the force is F = F/+ (7.00 N)5 - (6.70 N) and the corresponding torque about the origin is(6.10 Nm)i + (3.00 Nm)j + (-1.60 Nm). Determine Fx N
The direction of torque vector is perpendicular to the plane containing r and force, in the direction given by the right hand rule. The value of Fx is 0.522 N.
Position vector, r = 7 = (2.00 mi - (3.00 m)ſ + (2.00 m))Force vector, F = (7.00 N)5 - (6.70 N)Torque vector, τ = (6.10 Nm)i + (3.00 Nm)j + (-1.60 Nm)The equation for torque is given as : τ = r × FWhere, × represents cross product.The cross product of two vectors is a vector that is perpendicular to both of the original vectors and its magnitude is given as the product of the magnitudes of the original vectors times the sine of the angle between the two vectors.Finding the torque:τ = r × F= | r | | F | sinθ n, where n is a unit vector perpendicular to both r and F.θ is the angle between r and F.| r | = √(2² + 3² + 2²) = √17| F | = √(7² + 6.70²) = 9.53 sinθ = τ / (| r | | F |)n = [(2.00 mi - (3.00 m)ſ + (2.00 m)) × (7.00 N)5 - (6.70 N)] / (| r | | F | sinθ)
By using the right hand rule, we can determine the direction of the torque vector. The direction of torque vector is perpendicular to the plane containing r and F, in the direction given by the right hand rule. Finding Fx:We need to find the force component along the x-axis, i.e., FxTo solve for Fx, we will use the equation:Fx = F cosθFx = F cosθ= F (r × n) / (| r | | n |)= F (r × n) / | r |Finding cosθ:cosθ = r . F / (| r | | F |)= [(2.00 mi - (3.00 m)ſ + (2.00 m)) . (7.00 N) + 5 . (-6.70 N)] / (| r | | F |)= (- 2.10 N) / (| r | | F |)= - 2.10 / (9.53 * √17)Fx = (7.00 N) * [ (2.00 mi - (3.00 m)ſ + (2.00 m)) × [( - 2.10 / (9.53 * √17)) n ] / √17= 0.522 NTherefore, the value of Fx is 0.522 N.
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An automobile traveling 76.0 km/h has tires of 70.0 cm diameter. (a) What is the angular speed of the tires about their axles? (b) If the car is brought to a stop uniformly in 39.0 complete turns of the tires, what is the magnitude of the angular acceleration of the wheels? (c) How far does the car move during the braking? (
(a) Angular speed: 60.3 rad/s
(b) Angular acceleration: 0.244 rad/s²
(c) Distance moved: 5182.4 meters
(a) To find the angular speed of the tires about their axles, we can use the formula:
Angular speed (ω) = Linear speed (v) / Radius (r)
First, let's convert the speed from km/h to m/s:
76.0 km/h = (76.0 km/h) * (1000 m/km) * (1/3600 h/s) ≈ 21.1 m/s
The radius of the tire is half of its diameter:
Radius (r) = 70.0 cm / 2 = 0.35 m
Now we can calculate the angular speed:
Angular speed (ω) = 21.1 m/s / 0.35 m ≈ 60.3 rad/s
Therefore, the angular speed of the tires about their axles is approximately 60.3 rad/s.
(b) To find the magnitude of the angular acceleration of the wheels, we can use the formula:
Angular acceleration (α) = Change in angular velocity (Δω) / Time (t)
The change in angular velocity can be found by subtracting the initial angular velocity (ω_i = 60.3 rad/s) from the final angular velocity (ω_f = 0 rad/s), as the car is brought to a stop:
Δω = ω_f - ω_i = 0 rad/s - 60.3 rad/s = -60.3 rad/s
The time (t) is given as 39.0 complete turns of the tires. One complete turn corresponds to a full circle, or 2π radians. Therefore:
Time (t) = 39.0 turns * 2π radians/turn = 39.0 * 2π rad
Now we can calculate the magnitude of the angular acceleration:
Angular acceleration (α) = (-60.3 rad/s) / (39.0 * 2π rad) ≈ -0.244 rad/s²
The magnitude of the angular acceleration of the wheels is approximately 0.244 rad/s².
(c) To find the distance the car moves during the braking, we can use the formula:
Distance (d) = Linear speed (v) * Time (t)
The linear speed is given as 21.1 m/s, and the time is the same as calculated before:
Time (t) = 39.0 turns * 2π radians/turn = 39.0 * 2π rad
Now we can calculate the distance:
Distance (d) = 21.1 m/s * (39.0 * 2π rad) ≈ 5182.4 m
Therefore, the car moves approximately 5182.4 meters during the braking.
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In 2022, a 25-year-old astronaut left Earth to explore the galaxy; her spaceship travels at 2.5×10 ^8 m/s. She will return in 2035 . About how old will she appear to be? Justify your answer with one or more equations. () Calculate the work function that requires a 410 nm photon to eject an electron of 2.0eV. (Hint: Look for the values of constants on the formula sheet.) () An electron is moving at 3.8×10 ^6 m/s. What wavelength photon would have the same momentum? ()
The wavelength of a photon with the same momentum as an electron moving at 3.8×10^6 m/s.
To determine how old the astronaut will appear to be upon her return in 2035, we need to account for the effects of time dilation due to her high velocity during space travel.
According to the theory of relativity, time dilation occurs when an object is moving relative to an observer at a significant fraction of the speed of light.
The equation that relates the time experienced by the astronaut (Δt') to the time measured on Earth (Δt) is given by:
Δt' = Δt / γ
where γ is the Lorentz factor, defined as:
γ = 1 / sqrt(1 - v^2/c^2)
In this equation, v is the velocity of the astronaut's spaceship (2.5×10^8 m/s) and c is the speed of light (approximately 3×10^8 m/s).
To calculate the value of γ, substitute the values into the equation and evaluate it. Then, calculate the time experienced by the astronaut (Δt') using the equation above.
The difference in time between the astronaut's departure (2022) and return (2035) is Δt = 2035 - 2022 = 13 years. Subtract Δt' from the departure year (2022) to find the apparent age of the astronaut upon her return.
For the second question regarding the work function, the work function (Φ) represents the minimum energy required to remove an electron from a material. It can be calculated using the equation:
Φ = E_photon - E_kinetic
where E_photon is the energy of the photon and E_kinetic is the kinetic energy of the ejected electron.
In this case, the energy of the photon is given as 410 nm, which can be converted to Joules using the equation:
E_photon = hc / λ
where h is the Planck constant (6.626×10^-34 J·s), c is the speed of light, and λ is the wavelength in meters.
Calculate the energy of the photon and then substitute the values into the equation for the work function to find the answer.
For the third question regarding the wavelength of a photon with the same momentum as an electron moving at 3.8×10^6 m/s, we can use the equation that relates the momentum (p) of a photon to its wavelength (λ):
p = h / λ
Rearrange the equation to solve for λ and substitute the momentum of the electron to find the corresponding wavelength of the photon.
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A proton (charge +e, mass m.), a deuteron (charge +e, mass 2m), and an alpha particle (charge +2e, mass 4m,) are accel- erated from rest through a common potential difference AV. Each of the particles enters a uniform magnetic field B, with its velocity in a direction perpendicular to B. The proton moves in a circular path of radius r. In terms of r determine (a) the radius r of the circular orbit for the deu- teron and (b) the radius r for the alpha particle. α
The radius of the circular orbit for the deuteron and the alpha particle can be determined in terms of the radius r of the circular orbit for the proton.
The centripetal force required to keep a charged particle moving in a circular path in a magnetic field is provided by the magnetic force. The magnetic force is given by the equation F = qvB, where q is the charge of the particle, v is its velocity, and B is the magnetic field strength.
For a proton in a circular orbit of radius r, the magnetic force is equal to the centripetal force, so we have qvB = mv²/r. Rearranging this equation, we find that v = rB/m.
Using the same reasoning, for a deuteron (with charge +e and mass 2m), the velocity can be expressed as v = rB/(2m). Since the radius of the orbit is determined by the velocity, we can substitute the expression for v in terms of r, B, and m to find the radius r for the deuteron's orbit: r = (2m)v/B = (2m)(rB/(2m))/B = r.
Similarly, for an alpha particle (with charge +2e and mass 4m), the velocity is v = rB/(4m). Substituting this into the expression for v, we get r = (4m)v/B = (4m)(rB/(4m))/B = r.
Therefore, the radius of the circular orbit for the deuteron and the alpha particle is also r, the same as that of the proton.
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A ball of mass 0.606 kg moving east (+z direction) with a speed of 3.84 m/s collides head-on with a 0.303 kg ball at rest Assume that the collision is perfectly elastic Part A What is be the speed of the 0.606-kg ball after the collision?
The speed of the 0.606-kg ball after the collision is 2.56 m/s in the opposite direction.
Mass of the first ball (m₁) = 0.606 kg
Mass of the second ball (m₂) = 0.303 kg
Initial speed of the first ball (u₁) = 3.84 m/s
Initial speed of the second ball (u₂) = 0 m/s
The collision is said to be perfectly elastic. Therefore, kinetic energy is conserved.
Let's calculate the initial momentum and the final momentum of the balls using the principle of conservation of momentum.Initial momentum, P = m₁u₁ + m₂u₂
After the collision, the balls move in opposite directions. Let the velocity of the first ball be v₁ and that of the second ball be v₂. Then the final momentum, P' = m₁v₁ - m₂v₂
According to the law of conservation of momentum:
P = P' => m₁u₁ + m₂u₂ = m₁v₁ - m₂v₂
Therefore,
v₁ = [(m₁ - m₂)/(m₁ + m₂)]u₁ + [2m₂/(m₁ + m₂)]u₂v₂ = [2m₁/(m₁ + m₂)]u₁ + [(m₂ - m₁)/(m₁ + m₂)]u₂
Substituting the given values, we get:
v₁ = [(0.606 - 0.303)/(0.606 + 0.303)] × 3.84 + [2 × 0.303/(0.606 + 0.303)] × 0v₁ = 2.56 m/s
v₂ = [2 × 0.606/(0.606 + 0.303)] × 3.84 + [(0.303 - 0.606)/(0.606 + 0.303)] × 0v₂ = 1.28 m/s
Therefore, the speed of the 0.606-kg ball after the collision is 2.56 m/s in the opposite direction.
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A CONCAVE lens has the same properties as a CONCAVE mirror.
A. true
B. False
The Given statement "A CONCAVE lens has the same properties as a CONCAVE mirror" is FALSE because A concave lens and a concave mirror have different properties and behaviors.
A concave lens is thinner at the center and thicker at the edges, causing light rays passing through it to diverge. It has a negative focal length and is used to correct nearsightedness or to create virtual images.
On the other hand, a concave mirror is a reflective surface that curves inward, causing light rays to converge towards a focal point. It has a positive focal length and can produce both real and virtual images depending on the location of the object.
So, a concave lens and a concave mirror have opposite effects on light rays and serve different purposes, making the statement "A concave lens has the same properties as a concave mirror" false.
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How is momentum conserved when a large rolling boulder hits a smaller rolling boulder? Responses Momentum is not conserved when the two boulders collide. , Momentum is not conserved when the two boulders collide., , The smaller boulder had no momentum before the collision, and afterward it has all the momentum. , The smaller boulder had no momentum before the collision, and afterward it has all the momentum., , The larger boulder transfers some of its momentum to the smaller boulders, but it keeps going forward, too. , The larger boulder transfers some of its momentum to the smaller boulders, but it keeps going forward, too., , The larger boulder gives all of its momentum to the smaller boulder.
Explanation:
When a large rolling boulder hits a smaller rolling boulder, momentum is conserved. According to the law of conservation of momentum, the total momentum of a system remains constant if there are no external forces acting on it. In this case, the system consists of the two boulders.
During the collision, the larger boulder transfers some of its momentum to the smaller boulder, causing it to move forward. However, the larger boulder also continues to move forward with some of its original momentum. Therefore, the total momentum of the system before and after the collision remains the same.
remember that momentum is a vector quantity, meaning it has both magnitude and direction. The direction of momentum for each boulder will depend on their respective velocities and massez.
Answer:
The larger boulder transfers some of its momentum to the smaller boulder, but it keeps going forward, too. Therefore, option 5 is the correct response.
Explanation:
According to the law of conservation of momentum, the total momentum of a closed system remains constant before and after the collision, as long as no external forces are acting on it. When a large rolling boulder collides with a smaller rolling boulder, conservation of momentum takes place in the system.
During the collision, the larger boulder transfers some of its momentum to the smaller boulder through the force of the impact. This transfer of momentum causes the smaller boulder to gain some momentum and start moving in the direction of the collision
However, the larger boulder also retains some of its momentum and continues moving forward after the collision. Since the larger boulder typically has greater mass and momentum initially, it will transfer some momentum to the smaller boulder while still maintaining its own forward momentum.
Therefore, in the collision between the large rolling boulder and the smaller rolling boulder, momentum is conserved as both objects experience a change in momentum.
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At resonance, the current through an RLC circuit is: \( 5.0 \mathrm{~A} \) Maximized Minimized Zero
The maximum current through an RLC circuit can be calculated using the following equation: I(max) = V/R, where V is the voltage applied across the circuit and R is the resistance of the circuit. Therefore, the answer is maximized.
An RLC circuit is an electrical circuit containing a resistor, an inductor, and a capacitor, which are the three most commonly used electronic components. When a sinusoidal voltage is applied to an RLC series circuit, an alternating current (AC) flows through it.
The current through an RLC circuit at resonance is maximized. Resonance can be described as the point at which the inductive reactance of a coil is equal to the capacitive reactance of a capacitor. At this point, the inductive reactance and capacitive reactance cancel out, resulting in a minimum impedance in the circuit and a maximum current flow.
The phase angle between the current and voltage in an RLC circuit at resonance is zero, indicating that they are in phase. At resonance, the RLC circuit's current is determined solely by the resistance of the circuit's resistor. The current in an RLC circuit at resonance is determined by the following equation:
I = V/R
Where, V is the voltage applied across the circuit, R is the resistance of the circuit, and I is the current flowing through the circuit. At resonance, the current through an RLC circuit is maximized.
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6 An infinitely long wire along the z-axis carries 10A current. Find the magnetic flux density at a distance of 5m from the wire. 7 A 10 m long wire is aligned with the z-axis and is symmetrically placed at the origin Find the matio fold at filmoint/ 5)(ii) poi
6) Magnetic flux density at a distance of 5m from an infinitely long wire carrying 10A current can be calculated as follows;Magnetic field strength is directly proportional to the current. Therefore, we will use Ampere’s circuital law to calculate the magnetic flux density.
Let’s consider a circular path with radius r = 5m and let it be parallel to the wire. According to Ampere’s circuital law, [tex]∮.=enclosed≡I[/tex] Ampere’s circuital law where H is the magnetic field strength, I is the current and I enclosed is the current enclosed by the path.Now, we can find the H field strength by integrating along a circle of radius 5 m, we have, H = (10/2πr) T where T is the Tesla.
Therefore,
[tex]H = B/μ0 = [8/√2 x 10^-7]/[4π x 10^-7][/tex]
Tesla [tex]= 2/√2π Tesla = π Tesla/√2.[/tex]
Therefore, magnetic flux density at a distance of 5m from the infinitely long wire carrying 10A current is [tex]8π x 10^-7[/tex]Tesla. Magnetic field strength at a point P at a distance of 5m from the origin is π Tesla/√2.
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A long solenoid has n = 4000 turns per meter and carries a current given by I(t) = 50 (1e-1.6t) Where I is in Amperes and t is in seconds. Inside the solenoid and coaxial with it is a coil that has a radius of R = 2 cm and consists of a total N = 3500 turns of conducting wire. n turns/m ******************®®®® R O ooooooo oooooooo N turns What EMF (in Volts) is induced in the coil by the changing current at t = 1.5 s?
At t = 1.5 s, the changing current in the solenoid induces an EMF (electromotive force) of approximately 7.91 V in the coaxial coil.
To calculate the induced EMF in the coil, we need to determine the magnetic flux through the coil and then apply Faraday's law of electromagnetic induction.
1. Magnetic flux through the coil:
The magnetic flux through the coil is given by the equation Φ = B · A · N, where B is the magnetic field, A is the area of the coil, and N is the number of turns.
The magnetic field inside a solenoid is given by the equation B = μ₀ · n · I, where μ₀ is the permeability of free space, n is the number of turns per meter, and I is the current flowing through the solenoid.
Substituting the given values, the magnetic field inside the solenoid is B = (4π × 10⁻⁷ T·m/A) · (4000 turns/m) · [50 (1e^(-1.6 × 1.5)) A].
The area of the coil is A = π · R², where R is the radius of the coil.
2. EMF induced in the coil:
According to Faraday's law of electromagnetic induction, the induced EMF in the coil is given by the equation ε = -dΦ/dt, where ε is the induced EMF and dΦ/dt is the rate of change of magnetic flux.
To find the rate of change of magnetic flux, we need to differentiate the magnetic flux equation with respect to time. Since the magnetic field inside the solenoid is changing with time, we also need to consider the time derivative of the magnetic field.
Finally, substitute the values at t = 1.5 s into the derived equation to calculate the induced EMF in the coil.
By following these steps, we find that at t = 1.5 s, the changing current in the solenoid induces an EMF of approximately 7.91 V in the coaxial coil.
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Taking into account the following figure, the cart of m2=500 g on the track moves by the action of the weight that is hanging with mass m1=50 g. The cart starts from rest, what is the distance traveled when the speed is 0.5 m/s? (Use: g= 9.78 m/s2).. Mark the correct answer.
a. 0.10 m
b. 0.14 m
c. 0.09 m
d. 0.16 m
The distance traveled when the speed is 0.5 m/s is approximately 0.16 m.
To solve this problem, we can use the principle of conservation of mechanical energy. The potential energy of the hanging weight is converted into the kinetic energy of the cart as it moves.
The potential energy (PE) of the hanging weight is given by:
PE = m1 * g * h
where m1 is the mass of the hanging weight (50 g = 0.05 kg), g is the acceleration due to gravity (9.78 m/s^2), and h is the height the weight falls.
The kinetic energy (KE) of the cart is given by:
KE = (1/2) * m2 * v^2
where m2 is the mass of the cart (500 g = 0.5 kg) and v is the speed of the cart (0.5 m/s).
According to the principle of conservation of mechanical energy, the initial potential energy is equal to the final kinetic energy:
m1 * g * h = (1/2) * m2 * v^2
Rearranging the equation, we can solve for h:
h = (m2 * v^2) / (2 * m1 * g)
Plugging in the given values, we have:
h = (0.5 * (0.5^2)) / (2 * 0.05 * 9.78)
h ≈ 0.16 m
Therefore, the distance traveled when the speed is 0.5 m/s is approximately 0.16 m. The correct answer is (d) 0.16 m.
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An RLC circuit has a capacitance of 0.47μF. a) What inductance will produce a resonance frequency of 96MHz ? It is desired that the impedance at resonance be one-third the impedance at 27kHz. What value of R should be used to obtain this result?
An RLC circuit has a capacitance of 0.47 μF. We need to find the inductance and value of R.
The solution to it is explained below: Given data:
Capacitance (C) = 0.47 μF
Resonance frequency (f) = 96 MHz
Impedance at resonance (Z) = Impedance at 27 kHz/3
The resonance frequency can be found using the formula:
f = 1 / 2π√(LC)
The above formula is known as the answer and is used to find out the value of inductance (L). So, rearranging the formula we get:
L = (1/4π²f²C)
L = (1/4π²×96×10⁶ ×0.47 ×10⁻⁶)
L = 41.49 μH
So, the inductance value is 41.49 μH.
Impedance at resonance can be determined as:
Z = √(R²+(Xl - Xc)²)
Here, Xl is the inductive reactance and Xc is the capacitive reactance at the resonant frequency. At resonance,
Xl = Xc,
so Xl - Xc = 0
Therefore, Z = R
We know that impedance at resonance (Z) should be one-third the impedance at 27 kHz.
Hence: Z = RZ₁
Z = R/3
Where, Z₁ is the impedance at 27 kHz So, R = 3 Z₁
Now, the conclusion is the formula of L and the value of R that satisfies the given conditions.
L = 41.49 μH
R = 3 Z₁.
The answer to the question is as follows inductance value is 41.49 μH and R = 3 Z₁.
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