write an Conceptual/Numerical applications based on key concepts of
Gravity and experience of weightlessness (300 words)
summary of Gravity and experience of weightlessness (150
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The initial velocity's horizontal component is 3 m/s and the vertical component is 6 m/s. The maximum height from the ground (in meters) reached by the cannonball is 0.459 m.

If a cannonball is fired from the ground with an initial velocity of 3 m/s for the horizontal component and 6 m/s for the vertical component.

Horizontal component vx = 4 m/s

Vertical component vy = 3 m/s

From.the relation

Maximum height h max = vy²/2g

h max = 3 ²/2 × 9.8= 0.459 m

Thus, the maximum height from the ground reached by the cannonball is 0.459 m.

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The equivalent resistance of the entire circuit is 1530.2 Ω and the potential across resistor R is 12V. The current across through resistors is 0.019 A, 2.857 A,0.02 A, and 0.038 A and the power dissipated by resistor R is 0.228 W.

Given information,

Resistors,

Rz =622 Ohm

R₂ = 4.2 Ohm

R₃ = 592 Ohm

R₄ = 312 Ohm

Potential difference, V =12 V

a) The equivalent resistance of the entire circuit,

The resistors are connected in series,

R = Rz + R₂ + R₃ + R₄

R = 622 + 4.2 + 592 + 312

R = 1530.2 Ω

Hence, the equivalent resistance of the entire circuit is 1530.2 Ω.

b)  The potential across the resistor is 12V.

The current across through resistors can be determined using Ohm's law,

V= IR

I = V/R

First: For Rz

I = 12/622

I = 0.019 A

Second: For R₂,

I = 12/4.2

I = 2.857 A

Third: For R₃,

I= 12/592

I = 0.02 A

Fourth: For R₄,

I = 12/312

I  = 0.038 A

Hence, the current across through the resistors are 0.019 A, 2.857 A,0.02 A, and 0.038 A

c) The potential across the resistor is 12V.

The current across through resistors can be determined using Ohm's law,

V= IR

I = V/R

First: For Rz

I = 12/622

I = 0.019 A

Second: For R₂,

I = 12/4.2

I = 2.857 A

Third: For R₃,

I= 12/592

I = 0.02 A

Fourth: For R₄,

I = 12/312

I  = 0.038 A

Hence, the current across through the resistors are 0.019 A, 2.857 A,0.02 A, and 0.038 A

d) The power dissipated by resistor R1 ,

Power = voltage × current

P = VI

P = 12 × 0.019 Ω  

P = 0.228 W

Hence, the power dissipated by resistor R is 0.228 W.

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Substance A is made up of 30 wt% water and the remaining sand. The sand is practically pure SiO2 with a density of 165 lb/ft³. To determine the density of substance A in lb/ft³, we will use the formula;

The remainder sand and 30% water make up substance A. The sand has a density of 165 lb/ft3 and is essentially pure SiO2. We will apply the formula to determine the density of material A in lb/ft3;

{density of the substance} = {density of the sand} × {fraction of the sand} + {density of the water} × {fraction of the water}.

First, we need to determine the density of water. At room temperature, the density of water is about 62.4 lb/ft³. Therefore, we can calculate the density of substance A as follows:

{density of the substance A} = 165 × (1 - 0.30) + 62.4 × 0.30

= 115.5 + 18.72

= 134.22 lb/ft³

Thus, the density of substance A is 134.22 lb/ft³.

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For an A.C circuit the peak current is 9.899 Amps, the power dissipated is 587.9 W, and, the peak voltage drop is 118.7 V respectively.

Given information,

RMS current, Ir = 7.0 Amps

Resistance, r = 12 Ohm

a) The peak current, Ip

Ir = Ip/√2

7.0 = Ip/√2

Ip = 7.0×√2

Ip = 9.899Amps

Hence, the peak current is 9.899Amps.

b) Power dissipated is,

P = 1/2 Ip² ×r

P = 1/2×96.04×12

P = 587.9 W

Hence, the power dissipated is 587.9 W.

c) Peak voltage drop, V

V = Ip ×r

V = 9.899 × 12

V = 118.7 V

Hence, the Peak voltage drop is 118.7 V.

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The potential energy of the book is 11.025 joules.

The potential energy of a book resting on the edge of a desk 0.75 m above the floor can be calculated using the formula:

PE = mgh,

where m is the mass of the book, g is the acceleration due to gravity, and h is the height of the book above the floor.

the mass of the book is 1.5 kg

the height is 0.75 m

the acceleration due to gravity is 9.8 m/s²,

we can substitute these values into the formula and solve for the potential energy.

PE = mgh

PE = (1.5 kg) x (9.8 m/s²) x (0.75 m)

PE = 11.025 J

Therefore, the potential energy of the book is 11.025 joules.

The potential energy of an object is defined as the energy that an object possesses due to its position relative to other objects in its surroundings.

In this case, the book has potential energy because it is positioned above the floor, and this position gives it the potential to do work when it falls to the ground.

The greater the height of the book above the ground, the greater its potential energy, and the greater the work that can be done when it falls.

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Therefore, the required focal length of the eyepiece for the compound microscope is approximately 2.42 cm.

To determine the required focal length of the eyepiece for a compound microscope with a desired overall magnification, we can use the formula for the total magnification of a compound microscope:

Total Magnification = Magnification of Objective × Magnification of Eyepiece

Given:

Desired Overall Magnification = 136x

Magnification of Objective = 12.0x

Let's denote the magnification of the eyepiece as M(eyepiece) and the focal length of the eyepiece as f(eyepiece).

Using the formula for total magnification:

Total Magnification = Magnification of Objective ×Magnification of Eyepiece

136x = 12.0x × M(eyepiece)

Now, let's solve for the magnification of the eyepiece:

M(eyepiece) = Total Magnification / Magnification of Objective

M(eyepiece) = 136x / 12.0x

M(eyepiece) = 11.33

The magnification of the eyepiece should be approximately 11.33.

We can also relate the magnification of the eyepiece to the focal length of the eyepiece using the formula:

Magnification of Eyepiece = (25 cm / f(eyepiece)) + 1

Substituting the known value of the magnification of the eyepiece:

11.33 = (25 cm / f(eyepiece)) + 1

Solving for the focal length of the eyepiece:

(25 cm / f(eyepiece)) = 11.33 - 1

(25 cm / f(eyepiece)) = 10.33

f(eyepiece) = 25 cm / 10.33

Calculating the value:

f(eyepiece) ≈ 2.42 cm

Therefore, the required focal length of the eyepiece for the compound microscope is approximately 2.42 cm.

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The annual depreciation charge using the straight-line method can be calculated as follows:First, calculate the depreciation value.

It's important to note that salvage value is the value of the asset at the end of its useful life period. Here's how you can calculate the depreciation value using the straight-line method:Initial cost of the asset - Salvage Value / Useless life periodThe initial cost of the asset is $50 million. The salvage value is 10% of the initial cost of the asset. 10% of $50 million is $5 million. The useless life period is ten years.Now you can plug in the numbers and calculate the annual depreciation value:$50,000,000 - $5,000,000 / 10 years= $4,500,000 / yearThe annual depreciation charge using the straight-line method is $4.5 million.

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A 55.9-g ice cube is initially at 0.0°C, then the change in entropy of the ice cube when it melts completely is approximately 68.05 J/K, and the change in entropy of the environment is approximately -68.05 J/K.

(a) Here, the change in entropy of the ice cube:

ΔS = Q / T

where Q=  heat transferred ,T = temperature.

The heat transferred (Q) is the latent heat of fusion (L) multiplied by the mass of the ice cube (m): The expression is explained below,

Q = L × m

the given values is calculated below:

L = 3.33 × [tex]10^5[/tex] J/kg m = 55.9 g = 0.0559 kg

Q = (3.33 ×  [tex]10^5[/tex] J/kg) × (0.0559 kg)

Calculating the value of Q:

Q ≈ 1.86027 × [tex]10^4[/tex] J

The temperature (T) in this case is the melting point of ice, which is 0.0°C or 273.15 K.

Now, one can calculate the change in entropy:

ΔS = Q / T = (1.86027 ×[tex]10^4[/tex] J) / (273.15 K) ≈ 68.05 J/K

Hence, the change in entropy of the ice cube when it melts completely is approximately 68.05 J/K.

(b) Change in entropy of the environment: The change in entropy of the environment can be calculated using the formula:

ΔS_env = -ΔS_cube

Since the entropy change of the ice cube is positive (it increases), the entropy change of the environment will be negative (it decreases).

Therefore, the change in entropy of the environment is approximately -68.05 J/K.

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The correct answer is: Mendel's laws of inheritance imply that p1 = 9/16,p2 = 3/16,p3 = 3/16, andp4 = 1/16

Mendel’s laws of inheritance implies that if the dominant gene and the recessive gene are present, then the dominant gene is expressed while the recessive gene is hidden.

The  four possible phenotypes:

Total Number of plants = 1609

Mendel's laws of inheritance imply that there are four possible phenotypes:

(1) tall cut-leaf, (2) tall potato-leaf, (3) dwarf cut-leaf, and (4) dwarf potato-leaf.

Mendel’s first law of segregation states that an organism contains two alleles for each trait. The alleles separate during the formation of gametes. The egg or sperm gets only one of the two alleles that are present in the organism.

The genotype and phenotype ratios obtained in the F2 generation can be predicted using the Punnett square.

When the Punnett square is used, it can be predicted that the offspring of P1 generation will have the following genotype and phenotype ratios:9/16 tall cut-leaf tomatoes (P1)3/16 tall potato-leaf tomatoes (P2)3/16 dwarf cut-leaf tomatoes (P3)1/16 dwarf potato-leaf tomatoes (P4)

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If a circuit has a resistor with a resistance of 15.0 2, and the power into the resistor is 0.6 Watts, and the voltage across the resistor is 3.0 volts then the current can be calculated using Ohm's law and the calculated value comes out to be b. 0.2 A.

Given to us is

Resistance (R) = 15.0 Ω

Power (P) = 0.6 Watts

Voltage (V) = 3.0 volts

To calculate the current through the resistor, we can use Ohm's Law, which states that the current (I) flowing through a resistor is equal to the voltage (V) across the resistor divided by the resistance (R).

We can use the formula for power, which is:

Power (P) = (Voltage (V))² / Resistance (R)

Rearranging the formula to solve for voltage:

Voltage (V) = sqrt(Power (P) × Resistance (R))

Plugging in the given values:

Voltage (V) = sqrt(0.6 × 15.0)

Voltage (V)  = sqrt(9)

Voltage (V) = 3.0 volts

Now, we can use Ohm's Law to calculate the current:

Current (I) = Voltage (V) / Resistance (R)

Current (I) = 3.0 / 15.0

Current (I) = 0.2 A

Therefore, the current through the resistor is 0.2 A. The correct answer is b. 0.2 A.

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The force on Charge 3 can be estimated using Coulomb's law as the vector sum of the forces attributable to Charges 1 and 2. The distance between Charge 3 and Charge 1 is around 6.403 m.

The distance between Charge 3 and Charge 2 is about 5 m. Using the charge values and distances, the magnitude of the force due to Charge 1 on Charge 3 may be calculated.

The amount of the force due to Charge 2 on Charge 3 can be estimated in the same way. Adding the vector forces yields the net force on Charge 3. Trigonometry can be used to calculate the magnitude and direction of the net force.

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The acceleration of the proton is approximately 862 x 10⁶ m/s². Therefore, option B is correct.

Given information:

Electric field strength (E) = 9 NC⁻¹

Initial velocity (v₀) = 0 m/s

Final velocity (v) = 1.2 x 10⁶ m/s

To find the acceleration of the proton, the equation for the acceleration experienced by a charged particle in an electric field can be used.

The acceleration (a) of a charged particle in an electric field is given by the equation:

a = [tex]\frac{q \times E}{m}[/tex]

In the case of a proton, the charge (q) is the elementary charge, e, which is approximately 1.602 x 10⁻¹⁹. The mass (m) of a proton is approximately 1.673 x 10⁻²⁷ kg.

a = [tex]\frac{q \times E}{m}[/tex]

a = [tex]\frac{(1.602 \times 10^{-19} C \times 9 NC^{-1})}{(1.673 \times 10^{-27} kg)}[/tex]

Simplifying the calculation:

a = [tex]\frac{(1.442 \times 10^{-18} C N)}{(1.673 \times 10^{-27} kg)}[/tex]

a ≈ 862 x 10⁶ m/s²

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Radio telescopes cannot simply scan the skies looking for signals, because they receive radio waves of low intensity from outer space, and these waves are mixed up with terrestrial interference such as radio or television signals. These radio waves, mostly of frequencies ranging between 10 MHz and 100 GHz, are extremely weak by the time they reach Earth because they have been traveling for millions of years through space.

Radio telescopes must be highly sensitive in order to detect these low-intensity radio waves. Radio telescopes are designed to minimize terrestrial interference by being placed in isolated areas far from civilization. To capture faint signals, radio telescopes must be very large in size. Many radio telescopes are composed of several large antennae connected together to form an interferometer. This type of design offers a much larger detection area compared to a single large antenna.Astronomers use computer algorithms and powerful computer processors to analyze radio signals. The signals received by radio telescopes can be processed to produce an image of the source of the signal, such as a star or galaxy. These radio images can provide valuable information about the properties of the celestial bodies they detect.

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Exchanges ENERGY with surroundings but not MATTER is a) a closed system. A system that does not exchange energy or matter with its surroundings is known as an isolated system.

A cowari system is a system that exchanges both energy and matter with its surroundings. System that exchanges energy with the surroundings but not matter is a closed system. Closed system is an isolated system that exchanges energy with its surroundings but not matter. Therefore, the correct answer is A-3, B-1, C-2.

Matching Processes:

A) Bobaric process 2) Constant Temperature

B) bothermal process 4) Constant Volume

C) bochoric process 3) Heat neither added or removed

D) Adiabatic process 1) Constant Pressure

Therefore, the correct option is A-2, B-4, C-3, D-1.When a body A is in thermal equilibrium with a body B, and also separately with a body C, then B and C will be in thermal equilibrium with each other is True.Standard thermometric substance chosen is Solid. A thermometric substance is a material used to measure temperature. Liquid-in-glass thermometers, gas thermometers, resistance thermometers, and thermocouples are examples of thermometric devices. Solid thermometric substance is the standard substance for measuring temperature. Therefore, the correct answer is Solid.

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The direction of the momentum changed, but the momentum remained the same before and after the contact.

In Newtonian physics, an object's mass and velocity are combined to form momentum, more precisely linear momentum or translational momentum. It has both a magnitude and a direction, making it a vector quantity. A heavy object weighing 2.0kg that was already travelling away at a velocity of 0.5m/s was struck by a small mass with a weight of 0.5kg and a velocity of 2.0m/s. The contact caused the light mass to bounce back and slow down, while the heavier mass was propelled ahead as a result. The direction of the momentum changed, but the momentum remained the same before and after the contact.

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If two sinusoidal waves differ in phase by 50∘ the amplitude, of the resultant wave is given by 1.8 ym, hence option D is correct.

According to question:

Two sinusoidal waves have the same angular frequency the same amplitude ym, and travel in the same direction in the same medium.

The amplitude of the sinusoidal waves is ym

Difference in phase, ∅ = 25∘

Amplitude of the resultant wave is

A = 2 ym cos (∅/2)

Putting the values of ∅

A =  2 ym cos (50∘/2)

= 2 × cos (25∘)  ym

= 1.8 ym

Thus, the amplitude of the resultant wave is 1.8 ym

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The compass needle moved approximately 21.8 degrees.

To measure a potential difference across a circuit element, the voltmeter must be connected in series. Option D is correct.

To calculate how much the compass needle moved, we need to find the difference between the measured magnetic field and the Earth's magnetic field. The needle will align itself with the resultant magnetic field.

The difference in the magnetic field is given by:

ΔB = B_wire - B_earth

ΔB = (1.7 × 10⁽⁻⁵⁾T) - (2.7 × 10⁽⁻⁵⁾ T)

ΔB = -1.0 × 10⁽⁻⁵⁾ T

The angle through which the needle moves can be calculated using the equation:

θ = arctan(ΔB / B_earth)

Substituting the values:

θ = arctan((-1.0 × 10⁽⁻⁵⁾ T) / (2.7 × 10⁽⁻⁵⁾ T))

Calculating the arctan, we find:

θ = -21.8°

Since the angle is negative, we consider the absolute value:

θ = 21.8°

10. When measuring potential difference (voltage) across a circuit element, the voltmeter needs to be connected in series with the element. By connecting the voltmeter in series, it becomes part of the same path as the current flowing through the element, allowing it to measure the potential difference accurately.

Connecting the voltmeter in parallel would create a short circuit and disrupt the normal current flow, affecting the voltage measurement. Connecting the voltmeter perpendicular or single-ended does not provide the necessary connection to measure the potential difference accurately. Option D is correct.

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A loud, shrill whistle have high amplitude, high frequency, hence option A is correct.

Sound's frequency and amplitude are its characteristics. Modifying these characteristics alters how sound is heard by listeners.

Pitch, or the perception of a frequency of sound, is the quality of hearing a sound at a certain frequency.

High frequency is associated with high pitch, and high pitch is heard as having a harsh sound.

The intensity of a sound, or the amount of energy it has per unit area, is used to determine how loud it is. The intensity rises as the amplitude does. A loud sound hence has a larger density.

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With one bulb removed, the other light bulbs will become brighter because they receive a higher current.

Hence, the correct option is C.

In a parallel circuit, each component (light bulb) is connected to the same voltage source independently.

When you remove or unscrew one light bulb in a parallel circuit, the other light bulbs will continue to receive the same voltage as before. With one bulb removed, the total resistance in the circuit decreases, resulting in an increase in the current flowing through the remaining bulbs.

As a result, the other light bulbs will become brighter because they receive a higher current.

Hence, the correct option is C.

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The magnitude of the magnetic force in figure (b) is 3 N.

In the figure (b), the force acting on the particle is the magnetic force, and the direction of the magnetic force is given by the right-hand rule. According to the right-hand rule, we can use the right hand to find the direction of the magnetic force. If we point the thumb of our right hand in the direction of the velocity of the charged particle, the fingers will curl around to point in the direction of the magnetic field. As the charged particle is moving into the magnetic field, the magnetic force will act on the particle in the upward direction.Using the equation, Fm = Bqv, we can calculate the magnetic force that acts on the particle. Here, B is the magnetic field, q is the charge on the particle, and v is the velocity of the particle. Given the values of B, q, and v, we can substitute them into the equation and solve for Fm.Fm = Bqv = (0.1 T)(1.5 x 10^-6 C)(2 x 10^5 m/s) = 3 NTherefore, the magnitude of the magnetic force in figure (b) is 3 N.

The magnitude of the magnetic force in figure (b) is 3 N.

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Lithium ion produces the light with the highest energy among all the metal ions.

The metal ion that produces the light with the highest energy is the ion of Lithium. This is because the metal ion of Lithium is the one which is nearest to the nucleus and has the lowest electron orbital. The electrons of the Lithium ion are closer to the nucleus, so they need less energy to transition to the excited state. Lithium produces the highest-energy light of all the metal ions.

To answer this question, we need to determine which metal ion has the electrons closest to the nucleus and the lowest electron orbital. In this case, the answer is lithium. Because its electrons are closer to the nucleus, they need less energy to transition to the excited state, resulting in the highest-energy light. This is why lithium is the metal ion that produces the highest-energy light.

In conclusion, Lithium ion produces the light with the highest energy among all the metal ions.

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The ratio do/f of the object distance to the focal length of the mirror can be calculated using magnification, and the calculated ratio comes out to be is 1.

Given to us is

Image size (height) = 1/6 of object size (height)

We know that for a convex mirror, the image formed is virtual and reduced in size.

The magnification equation for a convex mirror is given by:

magnification (m) = -di/do

Since the image is 1/6 the size of the object, the magnification is:

m = -1/6

We also know that for a convex mirror, the focal length (f) is positive.

Using the magnification equation, we have:

-1/6 = -di/do

Simplifying, we find:

di = do/6

In the case of a convex mirror, the focal length (f) is positive.

Now we can determine the ratio do/f (object distance to focal length ratio) by dividing do by f:

do/f = do / f

Since the object distance (do) and focal length (f) are both positive, the ratio do/f is equal to 1.

Therefore, the ratio do/f for the given scenario is 1.

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Some depictions are Disclaimer and Contextualization, Educational Initiatives, Cultural Authenticity, and Representation.

To combat the negative cultural depictions present in some of its earlier films, Disney has taken several steps.

Disclaimer and Contextualization: Disney has chosen to provide disclaimers or contextualization when releasing certain films.

Educational Initiatives: Disney has also invested in educational initiatives to promote understanding and dialogue around the historical context and cultural sensitivities portrayed in their films.

Cultural Authenticity and Representation: In recent years, Disney has made efforts to ensure cultural authenticity and representation in their films.

These are the few steps that Disney take to combat the negative cultural depictions of some of its earlier films.

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A Slinky can be used to demonstrate how waves transmit both energy and information. In essence, waves are the transfer of energy from one location to another through the medium of a mechanical disturbance. The energy is transmitted through a medium in waves, which are classified based on their physical characteristics. There are two types of waves, transverse and longitudinal waves.

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The maximum allowable span length of the simply supported beam is 7.8 meters.

As per data,

A W 533 x 93 simply supported beam with a span of 7.8 m carries a uniformly distributed load of 52 kN/m throughout its length.

The beam has the following properties:

Ix = 0.000556 m

Fy = 248 MPa

Depth, d = 533 mm

Web thickness, ty = 10.2 mm

The beam is laterally supported over its entire length.

The allowable flexural stress is 0.66Fy, allowable shearing stress is 0.4Fy, and allowable deflection is L/360.

We need to calculate the maximum allowable span length, which can be obtained by checking the allowable deflection.

Let us use the formula for the maximum deflection:

δmax = (5/384) × (wL⁴) / (EI,)

Where,

w = uniformly distributed load [w = 52 kN/m]

L = span length

E = Modulus of Elasticity of Steel [E = 200 GPa (from the table)]

I = Moment of Inertia of the Beam [l = 0.000556 m].

Firstly, calculate the value of EI:

Let's calculate EI:

EI = E × I

   = 200 × 10⁹ Pa × 0.000556 m⁴

   = 62.88 × 10⁶ Nm²

Then, calculate the maximum allowable deflection.

δmax = (5/384) × (wL⁴) / (EI)

         = (5/384) × (52 × 10³ N/m × (7.8 m)⁴) / (62.88 × 10⁶ Nm²)

         = 3.1 mm

The maximum allowable deflection (δmax) is

δmax = 7.8/360

         = 0.0217 m

         = 21.7 mm.

Therefore, the allowable deflection is less than the maximum allowable deflection.

So, the calculated value is safe and acceptable.

Therefore, the maximum allowable span length of the simply supported beam is 7.8 meters.

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√( (43 W × 2 Yt) / (m Zv²) ) is the launch velocity v in power is needed to accelerate the same projectile from rest to a launch speed of Zv in a time of Yat.

The pace at which an object's location changes in relation to a frame of reference and time is what is meant by speed. Although it may appear sophisticated, velocity is just the act of moving quickly in one direction. Since it is a vector quantity, the definition of velocity requires both magnitude (speed) and direction. It has a metre per second SI unit. A body is considered to be accelerating if its velocity changes, either in magnitude or direction.

W = (1/2)mv²

P = W/t = (1/2)mv²/t

P = (1/2) mZv² / Yt

v = √( (43 W × 2 Yt) / (m Zv²) )

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The plane mirror is needed. The phenomenon is lateral inversion. The phenomenon responsible for it is left-right reversal. The height is 5cm. The length of the image is equidistant from the mirror surface which is 9.0cm.

1) A plane mirror is needed to obtain a virtual image of the same size as the object. Because A plane mirror is a flat, smooth reflective surface where the reflection occurs. When light rays strike a plane mirror, they bounce off it at the same angle at which they hit it, resulting in a reflection.

2) Name of the phenomenon is called lateral inversion.Because Lateral inversion occurs because the reflection in a plane mirror involves the reversal of the direction of light rays. When light rays strike the mirror and bounce off, they change direction but maintain the same angle of incidence.

3) The phenomenon responsible for the effect is lateral inversion or left-right reversal.When you sit in front of a plane mirror and write with your right hand, the image in the mirror shows it as if you are writing with your left hand. This occurs because the reflection in the mirror involves the reversal of the direction of light rays.

4) The image of the candle is formed 50 cm to the right of the mirror, and its height is also 5.0 cm.

To summarize:

The image is formed 50 cm to the right of the plane mirror.

The height of the image is 5.0 cm, which is the same as the height of the candle.

5) The length of the image of the pencil formed by the mirror is 10.0 cm. Both ends of the image are equidistant from the mirror surface.

1/f = 1/d + 1/d'

For the tip of the pencil lead:

d = 12.0 cm

1/d' = -1/12

d' = -12.0 cm

For the end of the eraser:

d = 21.0 cm

d' = -21.0 cm

Length of the image = |d' of eraser - d' of tip

Length of the image = |(-21.0 cm) - (-12.0 cm)|

Length of the image = 9.0 cm

Therefore, the length of the image of the pencil formed by the plane mirror is 9.0 cm.

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The required area of the column section (67.47 cm²) is smaller than the gross area (135 cm²) of the selected W14.

As per data,

Dead load (D) = 2224 kN,

Live load (L) = 3114 kN,

Column Length (L) = 731.52 cm.

Column is pinned at both ends and laterally supported in the weak direction at the one-third points of the total column length.

Fy = 345 MPa

To design a column to support the given loads using Load and Resistance Factor Design (LRFD) method, we use the formula;

Factored load = φD + φL

Factored load = 1.2D + 1.6L (For Strength Design)Where,φD is the dead load factor.

Where,

φL is the live load factor.

φ is the load factor (1.2D and 1.6L are the load factors for Strength Design).

Given the dead load and live load,

Factored load = 1.2(2224) + 1.6(3114)

Factored load = 7220.8 kN

The maximum factored load that a W14 column can support is;

Axial load capacity = φcPn

Axial load capacity = φcFyAg

Where, φc is the resistance factor [φc = 0.9] (for HSS sections), Fy is the yield strength of the steel, A is the cross-sectional area of the column, and Ag is the gross cross-sectional area of the column.

The lightest W14 section to be selected, Pn can be calculated by the following formula:

Pn = φcFyAg

The gross area, Ag of a W14 section can be obtained from the AISC manual as;

Ag = 135 cm² (from table 1-1, page 1-13, AISC manual).

For a W14 section;

W14 × 145 (Section depth = 14.7 cm) × 135 (Ag = 135 cm²)

The area A for the selected W14 column can be determined as;

7220.8/0.9 × 345 = 145 × A × 135

Solving for A;

A = 67.47 cm²

For W14 × 145 section;

Ag = 135 cm² and A = 67.47 cm²

The required area of the column section (67.47 cm²) is smaller than the gross area (135 cm²) of the selected W14 section.

Therefore, this section is adequate for the required design loads.

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The density of the fluid is [tex]\(\rho \approx 1300 \, \text{kg/m}^3\)[/tex].

Using Bernoulli's equation:

[tex]\(P_1 + \frac{1}{2} \rho V_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho V_2^2 + \rho g h_2\)[/tex]

given,

h2 - h1 = 2.0 m

v1 = speed of fluid at lower level = 4.6 m/sec

v2 = speed of the fluid at the top

Using the Continuity theorem, We know that the flow rate of fluid will be equal at both positions, So

Q1 = Q2

[tex]\(A_1 V_1 = A_2 V_2\)[/tex]

[tex]\(A_1 = \pi r_1^2\) and \(A_2 = \pi r_2^2\), where \(r_1 = \frac{6.0}{2} = 3.0\) cm and \(r_2 = 2.0\) cm[/tex]

So,

[tex]v2 = v1 A1/A2 = v1 r1^2/r2^2[/tex]

[tex]v2 = 4.6\times (3.0/2.0)^2\\= 10.35m/s\\P1 = 34 kPa\\= 84\times10^3 Pa\\P2 = 2.8kPa\\= 2.8\times10^3 Pa[/tex]

[tex]\rho[/tex] = density of fluid = ? kg/m^3

Using given values: of P1 and P2

[tex]\(P_1 + 0.5 \rho V_1^2 + \rho g h_1 = P_2 + 0.5 \rho V_2^2 + \rho g h_2\)[/tex]

[tex]\(0.5} \rho (V_2^2 - V_1^2) + \rho g (h_2 - h_1) = P_2\)[/tex]

[tex]\(\rho = \frac{{P_1 - P_2}}{{0.5 \cdot (V_2^2 - V_1^2) + g \cdot (h_2 - h_1)}}\)[/tex]

[tex]\(\rho = \frac{{84000 - 2800}}{{0.5 \cdot (10.35^2 - 4.6^2) + 9.81 \cdot 2.0}}\)[/tex]

[tex]\(\rho \approx 1300 \, \text{kg/m}^3\)[/tex]

Therefore, the density of the fluid is [tex]\(\rho \approx 1300 \, \text{kg/m}^3\)[/tex].

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