Liquid sodium can be used as a heat transfer fluid in some nuclear reactors due to its high thermal conductivity and low neutron absorptivity. Its vapor pressure is 40.0 torr at 633 degree C and an enthalpy of vaporization of 1.00x10^2 kJ/mol. To what temperature can it be heated if a maximum of 500 torr pressure is allowed in the system?

Answer:

75 rotations

Explanation:

f0 = 0, f = 3000 rpm = 50 rps, t = 3 s

(a) use first equation of motion for rotational motion

w = w0 + α t

2 x 3.14 x 50 = 0 + α x 3

α = 104.67 rad/s^2

(b) Let θ be the angular displacement

use second equation of motion for rotational motion

θ = w0 t + 1/2 α t^2

θ = 0 + 0.5 x 104.67 x 3 x 3

θ = 471.015 rad

The angle turn in one rotation is 2 π radian.

Number of rotation = 471.015 / (2 x 3.14) = 75 rotations

Answer:

63.6 %

Explanation:

TH = 1100 K , Tc = 400 k

Efficiency is given by

η = 1 - Tc / TH

η = 1 - 400 / 1100

η = 1 - 0.36

η = 0.636

η = 63.6 %

Answer:

v = 7121.3 m/s

Explanation:

As we know that the centripetal force for the space shuttle is due to gravitational force of earth due to which it will rotate in circular path with constant speed

so here we will have

[tex]\frac{mv^2}{r} = \frac{GMm}{r^2}[/tex]

here we know that

r = orbital radius = 6370 km + 1482 km

[tex]r = 7.852 \times 10^6 m[/tex]

also we know that

[tex]M = 5.97 \times 10^{24} kg[/tex]

now we will have

[tex]v^2 = \frac{(6.67 \times 10^{-11})(5.97 \times 10^{24})}{7.852 \times 10^6}[/tex]

[tex]v^2 = 5.07 \times 10^7[/tex]

[tex]v = 7121.3 m/s[/tex]

The speed of the space shuttle that orbited the earth at an altitude of 1482km  will be    [tex]V=7121.3\dfrac{m}{s}[/tex]

As we know that the centripetal force for the space shuttle is due to the gravitational force of the earth due to which it will rotate in a circular path with constant speed

so here we will have

[tex]\dfrac{mv^2}{r} = \dfrac{GMm}{r^2}[/tex]

[tex]V^2=\dfrac{GM}{r}[/tex]

here we know that

[tex]\rm r= orbital \ radius =6370+1482=7852 \ km[/tex]

[tex]r=7.852\times10^6\ m[/tex]

mass of earth [tex]M=5.97\times10^{24}\ kg[/tex]

Gravitational constant  [tex]G=6.67\times10^{-11}[/tex]

By putting all the values we get

[tex]V^2=\dfrac{(6.67\times10^{-11} )(5.97\times10^{24})}{7.852\times10^{6}}[/tex]

[tex]V^2=5.07\times10^7[/tex]

[tex]V=7121.3 \dfrac{m}{s}[/tex]

Thus the speed of the space shuttle that orbited the earth at an altitude of 1482km  will be    [tex]V=7121.3\dfrac{m}{s}[/tex]

Answer:

[tex]f_{min} = 574.3 Hz[/tex]

[tex]f_{max} = 604.5 Hz[/tex]

Explanation:

As per Doppler's effect of sound we know that when source and observer moves relative to each other then the frequency of sound observed by the observer is different from real frequency of sound.

As the source is moving here in this case so the frequency is given as

[tex]f = f_o\frac{v}{v\pm v_s}[/tex]

part a)

for lowest frequency we will have

[tex]f_{min} = 589(\frac{343}{343 + R\omega})[/tex]

here we know that

R = 54.6 cm

[tex]\omega = 16.1 rad/s[/tex]

now we have

[tex]f_{min} = 589(\frac{343}{343 + 0.546(16.1)})[/tex]

[tex]f_{min} = 574.3 Hz[/tex]

part b)

for maximum frequency we will have

[tex]f_{max} = 589(\frac{343}{343 - R\omega})[/tex]

here we know that

R = 54.6 cm

[tex]\omega = 16.1 rad/s[/tex]

now we have

[tex]f_{max} = 589(\frac{343}{343 - 0.546(16.1)})[/tex]

[tex]f_{max} = 604.5 Hz[/tex]

Answer:

explained

Explanation:

Yes, the heating of filament is what causes the light production (photon emission), and this heating is caused because of current in the light bulb

(H= i^2*R*t i=current, H= heat, t= time and R= resistance).But using constant current source is not a good idea because in constant current source resistance is very low that can cause short circuit and ultimately fusing it. Whereas in constant voltage source current adjusts itself and prevents fusing because of high resistance in the circuit.

Final answer:

Using a constant current source for light bulbs is not ideal because the filament's resistance changes as it heats up, which would lead to inefficient operation and potential damage. A constant voltage source is preferred as it adapts better to the changing resistance, ensuring a stable operation.

Explanation:

The heating of the filament in light bulbs, which leads to light production, is inherently linked to the current flowing through the filament. Using a constant current source instead of a constant voltage source is not advisable due to the changing resistance of the filament. As the filament heats up, its resistance increases significantly, which, according to Ohm's law (V=IR), would necessitate a higher voltage to maintain the constant current. Initially, when the light bulb is turned on and the filament is at a lower temperature, the required voltage to maintain a constant current would be lower. As the temperature and resistance rise, so would the required voltage to maintain that current. This varying voltage could lead to inefficient operation and potentially damage the bulb or reduce its lifespan. The constant voltage approach is preferred as it naturally adapts to the changing resistance of the filament, providing a more stable and predictable operation of the bulb.

Answer:

13 sec

Explanation:

Hello

by definition the acceleration is a vector derived magnitude that indicates the speed variation per unit of time

[tex]a=\frac{V_{f}-V_{i}}{t_{f}-t_{i}}\\a(t_{f}-t_{i})={V_{f}-V_{i}  }}\\t_{f}-t_{i}=\frac{V_{f}-V_{i}}{a} \\\\t_{f}=\frac{V_{f}-V_{i}}{a}+t_{i} \\\\[/tex]

Let

Vi=4.0 m/s

a=1.0 m/s2

Vf=17.0 m/s

ti= (0)sec

t2= unknown

[tex]t_{f}=\frac{V_{f}-V_{i} }{a}+t_{i}\\t_{f}=\frac{17-4 }{1}+0\\t_{f}=\ 13\ sec\\[/tex]

Answer: 13 sec

I hope it helps

Answer:

a) [tex]v_{max} = 2\ \textup{m/s}[/tex]

b) [tex]v_{avg} = 1\ \textup{m/s}[/tex]

c) Q = 1.256 × 10⁻³ m³/s

Explanation:

Given:

The velocity profile as:

[tex]u(r) = 2(1-\frac{r^2}{R^2} )[/tex]

Now, the maximum velocity of the flow is obtained at the center of the pipe

i.e r = 0

thus,

[tex]v_{max}=u(0) = 2(1-\frac{0^2}{R^2} )[/tex]

or

[tex]v_{max} = 2\ \textup{m/s}[/tex]

Now,

[tex]v_{avg} = \frac{v_{max}}{2}\ \textup{m/s}[/tex]

or

[tex]v_{avg} = \frac{2}}{2}\ \textup{m/s}[/tex]

or

[tex]v_{avg} = 1\ \textup{m/s}[/tex]

Now, the flow rate is given as:

Q = Area of cross-section of pipe × [tex]v_{avg}[/tex]

or

Q = [tex]\frac{\pi D^2}{4}\times v_{avg}[/tex]

or

Q = [tex]\frac{\pi 0.04^2}{4}\times 1[/tex]

or

Q = 1.256 × 10⁻³ m³/s

For fully developed laminar pipe flow in a circular pipe, the maximum velocity is 2 m/s, the average velocity is (4/3) m/s, and the volume flow rate is (16/3)π cm^3/s. The velocity profile equation and the formulas for maximum velocity, average velocity, and volume flow rate are explained in detail.

For fully developed laminar pipe flow in a circular pipe, the velocity profile is given by the equation u(r) = 2(1-r^2/R^2) in m/s, where R is the inner radius of the pipe. To find the maximum velocity, we need to substitute r = R into the equation. The maximum velocity u_max is then equal to 2(1-1^2/R^2), which simplifies to 2 m/s.

The average velocity can be found by integrating the velocity profile equation over the entire cross-sectional area of the pipe. The cross-sectional area is given by A = πR^2, so the average velocity v_avg is equal to (1/A) times the integral of 2(1-r^2/R^2) over the range r = 0 to r = R. Simplifying the integral and dividing by A, we get v_avg = (4/3) m/s.

The volume flow rate Q can be calculated by multiplying the cross-sectional area A by the average velocity v_avg. Using the given diameter of the pipe, we can find the radius R = 2 cm. Substituting the values into the equation Q = (πR^2)(4/3), we get Q = (16/3)π cm^3/s.

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Answer:

Displacement is 565.69 m at 45° west of north

Explanation:

Let north represent positive y axis and east represent positive x axis.

We have journey started from 600 N. Michigan Avenue, and walked 3 blocks toward north, 4 blocks toward west, and 1 block toward north to a train station.

3 blocks toward north = 300 j m

4 blocks toward west = -400 i m

1 blocks toward north = 100 j m

Total displacement = -400 i + 400 j m

Magnitude

     [tex]s=\sqrt{(-400)^2+400^2}=565.69m[/tex]

Direction,

     [tex]\theta =tan^{-1}\left ( \frac{400}{-400}\right )=135^0[/tex]

     Direction is 45° west of north.

Displacement is 565.69 m at 45° west of north

Answer:

(a) 1.468 km/s

(b) 129.5 m

Explanation:

Use the formula for the wavelength is given by

[tex]\lambda = \frac{h}{mv}[/tex]

mass of electron, m = 9.1 x 10^-31 kg

(a) λ = 4.94 x 10^-7 m

[tex]\lambda = \frac{h}{mv}[/tex]

[tex]v = \frac{6.6\times 10^{-34}}{9.1\times 10^{-31}\times 4.94\times 10^{-7}}[/tex]

v = 1.468 x 10^3 m/s = 1.468 km/s

(b) v = 5.60 x 10^6 m/s

[tex]\lambda = \frac{h}{mv}[/tex]

[tex]\lambda  = \frac{6.6\times 10^{-34}}{9.1\times 10^{-31}\times 5.60\times 10^{6}}[/tex]

λ = 129.5 m

Answer:

The number of turns is 64.

Explanation:

Given that,

Magnetic field = 0.050 T

Area of coil = 100 cm²

Frequency = 60 Hz

Output voltage emf= 12 V

We need to calculate the number of turns

Using formula of induced emf

[tex]emf =NAB\omega[/tex]

[tex]N=\dfrac{emf}{A\times B\times2\pi f}[/tex]

[tex]N=\dfrac{12}{0.01\times0.050\times2\times3.14\times60}[/tex]

[tex]N =63.6 = 64\ turns[/tex]

Hence, The number of turns is 64.

Answer:

You need 63.66 turns.

Explanation:

The number of turns of a magnetic field is given by the following formula:

[tex]N = \frac{V}{S*T*2\pi f}[/tex]

In which N is the number of turns, V is the maximum output voltage, S is the area of the rotating coil, in square meters and T is the measure of the magnetic field and f is the frequency.

In this problem, we have that:

Suppose that you wish to construct a simple ac generator having an output of 12 V maximum when rotated at 60 Hz. This means that [tex]V = 12[/tex] and [tex]f = 60[/tex].

A uniform magnetic field of 0.050 T is available. This means that [tex]T = 0.050[/tex].

If the area of the rotating coil is 100 cm2, how many turns do you need?

This means that [tex]S = 0.01[/tex]m². So:

[tex]N = \frac{V}{S*T*2\pi f}[/tex]

[tex]N = \frac{12}{0.01*0.05*120\pi}[/tex]

[tex]N = 63.66[/tex]

You need 63.66 turns.

Answer:

32 J

Explanation:

Power is V²/R = 20²/1500 = 4/15 . . . watts

In 120 seconds, the energy is ...

(4/15 J/s)×(120 s) = 32 J

In a 2-minute period 32 joules are dissipated as heat.

We calculate the power dissipation rate of the 1.5-kΩ resistor under a constant potential difference of 20- V, which is 0.27 Watts or 0.27 Joules per second. Multiplying this rate by the time interval of two minutes (or 120 seconds), we find the total energy dissipated as heat, which is 32.4 Joules.

To understand the amount of energy dissipated during the given time interval, we first need to determine the power of the 1.5-kΩ resistor. This can be done using the equation P = V²/R, where V is the voltage, and R is the resistance. For this case, we have P = (20 V) ² / (1.5 kΩ) = 0.27 Watts.

Since the power dissipation rate is 0.27 Watts, which is equivalent to 0.27 Joules per second, we can now find out the energy dissipated during the two-minute interval using the equation E = Pt. Here, t represents the time in seconds, so t = 2 minutes * 60 seconds/minute = 120 seconds. Therefore, substituting the known values, we get E = (0.27 J/s)(120 s) = 32.4 Joules.

So, this is the total energy dissipated as heat by the 1.5- kΩ resistor under a constant potential difference of 20- V during the two-minute interval.

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The heat needed to evaporate 31.5 g of boiling water is approximately 7,150 J.

To calculate the heat needed to completely evaporate water, we use the equation Q = m × ΔHvap, where Q is the heat, m is the mass, and ΔHvap is the heat of vaporization.

The heat of vaporization for water is approximately 40.7 kJ/mol. To find the heat for 31.5 g of water, we need to convert the mass to moles by dividing by the molar mass of water (18.015 g/mol).

Using the equation, Q = (31.5 g ÷ 18.015 g/mol) × (40.7 kJ/mol × 1000 J/kJ), the answer is approximately 7,150 J.

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Final answer:

To evaporate 31.5 g of boiling water at 100°C, 70,875 joules of heat are needed. This is determined by multiplying the mass by the heat of vaporization for water, which is 2,250 J/g.

Explanation:

To calculate the amount of heat required to completely evaporate 31.5 g of boiling water at 100°C, the heat of vaporization of water is needed. The heat of vaporization of water is 2,250 J/g. To find the total heat we will multiply the mass of the water by the heat of vaporization:

Heat required = mass × heat of vaporization

Heat required = 31.5 g × 2,250 J/g

Heat required = 70,875 J

Therefore, 70,875 joules of heat would be needed to completely evaporate 31.5 g of boiling water at 100°C.

Answer:

11.515 Joule

Explanation:

Volume of aluminium = V = 4.89×10⁻³ m³

Coefficient of volume expansion for aluminum = α = 69×10⁻⁶ /°C

Initial temperature = 19.1°C

Final temperature = 357°C

Pressure of air = 1.01×10⁵ Pa

Change in temperature = ΔT= 357-19.1 = 337.9 °C

Change in volume

ΔV = αVΔT

⇒ΔV = 69×10⁻⁶×4.89×10⁻³×337.9

⇒ΔV = 114010.839×10⁻⁹ m³

Work done

W = PΔV

⇒W = 1.01×10⁵×114010.839×10⁻⁹

⇒W = 11.515 J

∴ Work is done by the expanding aluminum is 11.515 Joule

Answer:

Gravitational force, [tex]F=1.003\times 10^{-11}\ N[/tex]

Explanation:

It is given that,

Mass of roast beef sandwich, m₁ = 0.05 kg

Mass of hungry tarantula, m₂ = 87 gm = 0.087 kg

Distance between two objects, d = 17 cm = 0.17 m

We need to find the gravitational force between them. It is given by :

[tex]F=G\dfrac{m_1m_2}{d^2}[/tex]

[tex]F=6.67\times 10^{-11}\times \dfrac{0.05\ kg\times 0.087\ kg}{(0.17\ m)^2}[/tex]

[tex]F=1.003\times 10^{-11}\ N[/tex]

So, the gravitational force between two objects is [tex]F=1.003\times 10^{-11}\ N[/tex]. Hence, this is the required solution.

Answer:

6.29 m/s option (A)

Explanation:

theta = 45 degree, H = 1.01 m

let v be the launch speed

Use the formula for the maximum height for the projectile

H = v^2 Sin^θ / 2g

1.01 = v^2 x Sin^2(45) / (2 x 9.8)

1.01 = 0.0255 v^2

v^2 = 39.59

v = 6.29 m/s

The initial velocity of the grasshopper is 6.29 m/s.

The Initial velocity of the grasshopper is calculated from the following kinematic equation.

[tex]H = \frac{v_0^2 sin^2 \theta}{2g}[/tex]

where;

[tex]v_0^2 = \frac{2gH}{sin^2\theta} \\\\v_0^2 = \frac{2 \times 9.8 \times 1.01 }{(sin45)^2} \\\\v_0^2 = 39.6\\\\v_0 = 6.29 \ m/s[/tex]

Thus, the initial velocity of the grasshopper is 6.29 m/s.

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Answer:

3A is the larger of the two currents.

Explanation:

Let the currents in the two wires be I₁ and I₂

given:

Magnitude of the electric field, B = 4.0μT = 4.0×10⁻⁶T

Distance, R = 10cm = 0.1m

Ratio of the current = I₁ : I₂ = 3 : 1

Now, the magnitude of a magnetic field at a distance 'R' due to the current 'I' is given as

[tex]B = \frac{\mu_oI}{2\pi R}[/tex]

Where [tex]\mu_o[/tex] is the magnitude constant = 4π×10⁻⁷ H/m

Thus, the magnitude of a magnetic field due to I₁ will be

[tex]B_1 = \frac{\mu_oI_1}{2\pi R}[/tex]

[tex]B_2 = \frac{\mu_oI_2}{2\pi R}[/tex]

given,

B = B₁ - B₂ (since both the currents are in the same direction and parallel)

substituting the values of B, B₁ and B₂

we get

4.0×10⁻⁶T =  [tex]\frac{\mu_oI_1}{2\pi R}[/tex] - [tex]\frac{\mu_oI_2}{2\pi R}[/tex]

or

4.0×10⁻⁶T =  [tex]\frac{\mu_o}{2\pi R}\times (I_1-I_2 )[/tex]

also

[tex]\frac{I_1}{I_2} = \frac{3}{1}[/tex]

⇒[tex]I_1 = 3\times I_2[/tex]

substituting the values in the above equation we get

4.0×10⁻⁶T =  [tex]\frac{4\pi\times 10^{-7}}{2\pi 0.1}\times (3 I_2-I_2)[/tex]

⇒[tex]I_2 = 1A[/tex]

also

[tex]I_1 = 3\times I_2[/tex]

⇒[tex]I_1 = 3\times 1A[/tex]

⇒[tex]I_1 = 3A[/tex]

Hence, the larger of the two currents is 3A

Answer:

[tex]F = 92.45 N[/tex]

Explanation:

As we know that the force between two charge particles is given by

[tex]F = \frac{kq_1q_2}{r^2}[/tex]

here we know that

[tex]q_1 = 3.55 \mu C[/tex]

[tex]q_2 = 5.45 \mu C[/tex]

now the distance between the two charges is

r = 4.34 cm

now from the formula of electrostatic force we will have

[tex]F = \frac{(9\times 10^9)(3.55 \mu C)(5.45 \mu C)}{0.0434^2}[/tex]

[tex]F = 92.45 N[/tex]

Answer:

[tex]F_{net} = 220.8 N[/tex]

Explanation:

It is pulled by three forces as given below

1. Jack pulls directly ahead of the donkey with a force of 61.3 N,

2. Jill pulls with 83.9 N in a direction 45° to the left, and

3. Jane pulls in a direction 45° to the right with 137 N.

Now net force directly in forward direction given as

[tex]F_x = 61.3 N + 83.9 cos45 + 137cos45[/tex]

[tex]F_x = 217.5 N[/tex]

Now similarly in perpendicular to this we have

[tex]F_y = 137 sin45 - 83 sin45 [/tex]

[tex]F_y = 38.2 N[/tex]

Now net force is given by them

[tex]F_{net} = \sqrt{F_x^2 + F_y^2}[/tex]

[tex]F_{net} = \sqrt{217.5^2 + 38.2^2}[/tex]

[tex]F_{net} = 220.8 N[/tex]

Answer:

Velocity is 1.73 m/s along 54.65° south of east.

Explanation:

Let unknown velocity be v, mass of billiard ball be m and east direction be positive x axis.

Here momentum is conserved.

Initial momentum = Final momentum

Initial momentum = m x 2i + m x (-1)i = m i

Final momentum = m x v + m x 1.41 j = mv + 1.41 m j

Comparing

mi = mv + 1.41 m j

v = i - 1.41 j

Magnitude of velocity

      [tex]v=\sqrt{1^2+(-1.41)^2}=1.73m/s[/tex]        

Direction,  

       [tex]\theta =tan^{-1}\left ( \frac{-1.41}{1}\right )=-54.65^0[/tex]             

Velocity is 1.73 m/s along 54.65° south of east.

Using the law of conservation of momentum, one can deduce the speed and direction of the first ball after the collision. It's found to be traveling east at 1 m/s.

The scenario described is an example of a two-dimensional collision. In such collision, the law of conservation of momentum applies both in the east-west direction (x-axis) and the north-south direction (y-axis).

With the given velocities before the collision, the total momentum in the x-axis before the collision is: momentum_east = mass*velocity = m*2.00 m/s, and momentum_west = m*(-1.00) m/s. Therefore, total momentum in the x-axis before the collision = m*2.00 m/s + m*(-1.00) m/s = m m/s.

After the collision, the first ball keeps moving in the east-west direction (since the second ball is deflected north), but we don't know its velocity, let's call it v1. Applying the conservation of momentum after the collision in the x-direction, we get: total momentum = m*v1 + 0 (since the second ball no longer moves in the east-west direction) = m m/s. From this, we can solve for v1 and find that v1 = 1 m/s east. Thus, the first billiard ball is traveling east at 1 m/s after the collision.

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Answer:

[tex]T_2 = 309 N[/tex]

[tex]T_1 = 489 N[/tex]

Explanation:

As we know that total tension in both the ropes is counter balancing the weight of scaffold and worker both

so here we will have

[tex]T_1 + T_2 = (m + M)g[/tex]

now we have

[tex]T_1 + T_2 = 198 + 600 = 798 N[/tex]

now we also know that net torque due to both tension force in the string with respect to the position of worker must be zero so that platform will remain in equilibrium and horizontal in position

so here we will have

[tex]T_1(1.12) + (198)(1.6 - 1.12) = T_2(3.2 - 1.12)[/tex]

[tex]T_1+ 84.86 = 1.86 T_2[/tex]

now from above two equations we will have

[tex](1.86 T_2 - 84.86) + T_2 = 798 [/tex]

[tex]T_2 = 309 N[/tex]

also we have

[tex]T_1 = 489 N[/tex]

Final answer:

To calculate the tension in each rope supporting a scaffold with a window washer, the principles of torques and equilibrium are applied. By considering the system's equilibrium, equations based on the torques created by the worker's weight, the scaffold's weight, and the tensions in the ropes can be set up and solved.

Explanation:

To solve for the tension in each rope when a window washer stands on a scaffold, we use the concept of torques and equilibrium. The entire system is in equilibrium, meaning the sum of torques around any pivot point is zero. Considering the window washer weighing 600 N and standing 1.12 m from one end of a 3.2 m scaffold that weighs 198 N, we choose the pivot point at one end of the scaffold for easier calculations.

Let T1 be the tension in the rope closest to the worker and T2 be the tension in the other rope. The force due to the worker creates a clockwise torque, and the force due to the scaffold creates a clockwise torque as well, whereas the tensions in the ropes create counterclockwise torques. The torques are calculated by multiplying the force by the distance from the pivot point.

To find T1 and T2, we set up the equilibrium condition for torques and solve the equations considering the weight of the worker and the scaffold acting at their respective centers of mass. The specific calculations require numerical values that depend on the distances given and the gravitational force.

Ultimately, by solving these equations, we find the tensions T1 and T2 that support the system in equilibrium.

Answer:

The final gauge pressure will be 1.74 atm

Explanation:

Assume air tire as an ideal gas, therefore, it is considered:

The expression to describe the ideal gas process is:

[tex]P.v=Rg.T[/tex]

Where v is the specific volume or the inverse of the density p:

[tex]P/p=Rg.T[/tex] (2)

Then the P and T are the absolute pressure and temperature respectively. Rg represents the particular gas constant for air, Rg is equal to 287 J/kg-K. Take into account that P and T must be expressed in Pascal and Kelvin respectively.

By reorganizing  the expression (2) as below is doing:

[tex]P/T=Rg.p[/tex]  

Can be noticed that the product Rg*p is constant therefore relation P/T will also be constant. For two different states 1 and 2 of the ideal gas, it follows:

[tex]P_{1}/T_{1}=P_{2}/T_{2}[/tex] (3)

Here, states 1 and 2 will represent the state before and after arriving in Alaska.

Note that the temperatures given are in °C so it must be converted:

[tex]T(K)=273+T(°C)[/tex]  

[tex]T_{1} (K)=273+35[/tex]  

[tex]T_{1}=308 K[/tex]  

[tex]T_{2} (K)=273-42[/tex]

[tex]T_{2}=231 K[/tex]

Note also that the pressure given is the gauge pressure therefore it must be expressed as absolute pressure:

[tex]Pa (Pa)=Patm(Pa)+Pg(Pa)[/tex]  

Where Patm is the atmosphere pressure and is equal to 101325 Pa then for Pg of 2.7 10^5 Pa:

[tex]Pa_{1} (Pa)= 101325 Pa+ 270000 Pa[/tex]  

[tex]Pa_{1} (Pa)= 371325 Pa[/tex]  

Solving equation (3) for pressure at state 2:

[tex]P_{2}=P_{1}*T_{2}/T_{1}[/tex]

For the temperatures and pressure calculated values:

[tex]P_{2}= 371325 Pa * \frac231K}/{308K}[/tex]

[tex]P_{2}= 278493.75 Pa [/tex]

As is required the gauge pressure and not the absolute pressure:

[tex]Pg (Pa)=Pa(Pa)-Patm(Pa)[/tex]  

[tex]Pg_{2} (Pa)= 278493.75 -101325(Pa)[/tex]  

[tex]Pg_{2} (Pa)= 177168 Pa[/tex]  

Finally re expressing the pressure in atm units:

[tex]P (atm)=P (Pa)*\frac{1 atm}{101325 Pa}[/tex]  

[tex]P_{2} (atm)=177168*\frac{1 atm}{101325 Pa}[/tex]  

[tex]P_{2} (atm)=1.74 atm[/tex]

Final answer:

Using Gay-Lussac's law, the gauge pressure of the car tires drops from 2.50 imes [tex]10^5 N/m^2[/tex] at 35.0°C to approximately 1.85 atm at -42.0°C, after converting the final pressure to atmospheres.

Explanation:

The question pertains to the pressure change of a car tire due to a temperature drop, which is a direct application of the Ideal Gas Law. To determine the new gauge pressure in the tires after the temperature drop from 35.0°C to -42.0°C, we can use the combined gas law which relates the pressure, volume, and temperature of a gas.

Since the volume of the tires is constant (the tire has not gained or lost any air), the combined gas law simplifies to Gay-Lussac's law:

P1/T1 = P2/T2

Here, P1 and T1 are the initial pressure and temperature, and P2 and T2 are the final pressure and temperature. We must convert the temperatures to kelvin, which is the absolute temperature scale used in gas law calculations:

T1(K) = 35.0 + 273.15 = 308.15 K
T2(K) = -42.0 + 273.15 = 231.15 K

We can now rearrange the equation to solve for P2:

P2 = P1 * (T2/T1)

Substituting the given values:

P2 = 2.50 × 105 N/m² * (231.15 K / 308.15 K)
P2 = 1.8725 × 105 N/m²

However, the question asks for the pressure in atmospheres. We convert using the conversion factor 1 atm = 1.013 × 105 N/m²:

P2(atm) = 1.8725 × 105 N/m² / 1.013 × 105 N/m²
P2(atm) \  1.85 atm

Therefore, the gauge pressure of the car tires when the temperature has dropped to -42.0°C is approximately 1.85 atm.

The capacitance of a capacitor is the ratio of the stored charge to its potential difference, i.e.

C = Q/ΔV

C is the capacitance

Q is the stored charge

ΔV is the potential difference

Rearrange the equation:

ΔV = Q/C

We also know the capacitance of a parallel-plate capacitor is given by:

C = κε₀A/d

C is the capacitance

κ is the capacitor's dielectric constant

ε₀ is the electric constant

A is the area of the plates

d is the plate separation

If we substitute C:

ΔV = Qd/(κε₀A)

We assume the stored charge and the area of the plates don't change. Then if we double the plate spacing, i.e. we double the value of d, then the potential difference ΔV is also doubled.

Answer:

1.77 x 10^-8 C

Explanation:

Let the surface charge density of each of the plate is σ.

A = 4 x 4 = 16 cm^2 = 16 x 10^-4 m^2

d = 2 mm

E = 2.5 x 10^6 N/C

ε0 = 8.85 × 10-12 C2/N ∙ m2

Electric filed between the plates (two oppositively charged)

E = σ / ε0

σ = ε0 x E

σ = 8.85 x 10^-12 x 2.5 x 10^6 = 22.125 x 10^-6 C/m^2

The surface charge density of each plate is ± σ / 2

So, the surface charge density on each = ± 22.125 x 10^-6 / 2

                                                                 = ± 11.0625 x 10^-6 C/m^2  

Charge on each plate = Surface charge density on each plate x area of each plate

Charge on each plate = ± 11.0625 x 10^-6  x 16 x 10^-4 = ± 1.77 x 10^-8 C

Answer:

Lifting force, F = 21240 N

Explanation:

It is given that,

Mass of the helicopter, m = 1800 kg

It rises with an upward acceleration of 2 m/s². We need to find the lifting force  supplied by its rotating blades. It is given by :

F = mg + ma

Where

mg is its weight

and "ma" is an additional acceleration when it is moving upwards.

So, [tex]F=1800\ kg(9.8\ m/s^2+2\ m/s^2)[/tex]

F = 21240 N

So, the lifting force supplied by its rotating blades is 21240 N. Hence, this is the required solution.

The lifting force supplied by the helicopter's blades is calculated using Newton's second law. It's the sum of the force needed to counteract the weight of the helicopter and the force needed to provide the upward acceleration. Therefore, the lifting force is 21240 N.

To calculate the lifting force, we need to use the principle of Newton's second law applied vertically, which says that Force equals mass times acceleration (F = ma). For a helicopter to rise, the upward force supplied by the rotating blades (Lift) must be greater than the downward force of gravity (Weight).

The weight of the helicopter is given by its mass times the gravitational acceleration, which is approximately 9.8 m/s^2. So, the weight of the helicopter is 1800 kg * 9.8 m/s^2 = 17640 N.

The total force needed for the helicopter to rise with an acceleration of 2 m/s^2 upwards is given by its mass times this acceleration, so that's 1800 kg * 2 m/s^2 = 3600 N.

Therefore, the lifting force supplied by the blades is the sum of the force needed to counteract the weight and the force needed to provide the upward acceleration. That is, 17640 N + 3600 N = 21240 N.

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Answer:

W = 2.158 eV

fo = 5.23 x 10^14 Hz

Explanation:

f = 7.9 x 10^14 Hz, Vo = 1.1 V

Let W be the work function.

Use the Einstein equation

Energy = W + eVo

hf = W + eVo

where, h is the Plank's constant and e be the electronic charge.

W = hf - eVo

W = (6.6 x 10^-34 x 7.9 x 10^14) - (1.6 x 10^-19 x 1.1)

W = 5.214 x 10^-19 - 1.76 x 10^-19 = 3.454 x 10^-19 J

W = (3.454 x 10^-19) / (1.6 x 10^-19) = 2.158 eV

Let fo be the cut off frequency

W = h fo

fo = W / h = (3.454 x 10^-19) / (6.6 x 0^-34) = 5.23 x 10^14 Hz

Answer:

The change in the area of the circle is [tex]8.88\times10^{-7}\ m^2[/tex]

Explanation:

Given that,

Radius = 0.250 cm

Temperature = 20.0°C

Final temperature =800.0°C

Coefficient of linear thermal expansion for lead[tex]\alpha =29.0\times10^{-6}\ /\°C [/tex]

We calculate the change in temperature,

[tex]\Delta T=800.0-20.0=780^{\circ}[/tex]

Now, We calculate the area of the disc

[tex]A = \pi r^2[/tex]

Put the value into the formula

[tex]A=3.14\times(2.5\times10^{-3})^2[/tex]

[tex]A =1.9625\times10^{-5}\ m^2[/tex]

We need to calculate the areal expansion

[tex]\Delta A=2\alpha\times A\times\Delta T[/tex]

[tex]\Delta A=2\times29.0\times10^{-6}\times1.9625\times10^{-5}\times780[/tex]

[tex]\Delta A=8.88\times10^{-7}\ m^2[/tex]

Hence, The change in the area of the circle is [tex]8.88\times10^{-7}\ m^2[/tex]

Final answer:

The change in area of the lead circular disc can be calculated using the formulas for linear thermal expansion. The change in radius is calculated using the coefficient of linear expansion and the change in temperature. The change in area is then calculated using the change in radius and the original radius.

Explanation:

To determine the change in area of the circular disc made of lead, we need to calculate the change in its radius using the coefficient of linear thermal expansion. The formula for linear thermal expansion is given by AL = a * L * AT, where AL is the change in length, a is the coefficient of linear expansion, L is the original length, and AT is the change in temperature.

In this case, we are interested in the change in radius, so we can use the formula AR = a * R * AT, where AR is the change in radius and R is the original radius.

Substituting the given values, we have:

AR = (29.0 × 10^-6/°C) * (0.250 cm) * (800.0 °C - 20.0 °C)

AR = 0.00145 cm

The change in area can be calculated using the formula AE = 2π * R * AR. Substituting the values, we have:

AE = 2π * (0.250 cm) * (0.00145 cm)

AE = 2.31 x 10^-3 cm²

Answer:

20.26 Nm

Explanation:

r = 1.23 m , m = 93.6 kg, w = 0, f0 = 369 rpm = 369 / 60 = 6.15 rps

w0 = 2 x 3.14 x 6.15 = 38.622 rad/s

t = 2.25 min = 2.25 x 60 = 135 second

Moment of inertia = 1/2 m r^2 = 0.5 x 93.6 x 1.23 x 1.23 = 70.8 kg m^2

use first equation of motion for rotational motion

w = w0 + α t

0 = 38.622 - α x 135

α = 0.286 rad/s^2

torque = moment of inertia x angular acceleration

Torque = 70.8 x 0.286 = 20.26 Nm

Power dissipated by a resistor = (current)² x (resistance).

0.25 W = (0.02 A)² x (resistance)

This resistor = (0.25 W) / (0.02 A)²

This resistor = 625 ohms

If we want it to dissipate 0.5 W, then

0.5 W = (current)² x (625 ohms)

Current = √(0.5/625)

Current = √(0.0008)

Current = 28.28 mA

============================

A slightly easier way:

Since the power is = I²R, it grows in proportion to (current)² .

We want to double the power dissipated, so we only need to increase the current by the factor of √2 .

(20 mA) x (√2) = 28.28 mA

Final answer:

Using the power dissipation formula, we calculate the resistance as 625 ohms. To dissipate 0.50 W, a current of sqrt(0.50 W / 625 ohms) is required, which equals 28 mA.

Explanation:

To determine how much current would be needed for a resistor to dissipate 0.50 W, we'll use the formula for electric power dissipation in a resistor which is P = I^2 R, where P is the power in watts, I is the current in amperes, and R is the resistance in ohms. In the given example, a resistor dissipates 0.25 W when a 20 mA (or 0.02 A) current passes through it. With P = I^2 R, we can derive R as R = P / I^2 which gives a resistance value through this initial condition.

The first step is to calculate the resistance of the resistor using the initial power dissipation value:

R = 0.25 W / (0.02 A)^2 = 0.25 / 0.0004 = 625 ohms.

Now, to find the current for 0.50 W power dissipation, we use

I = sqrt(P / R) = sqrt(0.50 W / 625 ohms) = sqrt(0.0008) = 0.0283 A or 28.3 mA.

Therefore, 28 mA of current is needed for the resistor to dissipate 0.50 W.

Answer:

Option B

Explanation:

The net emf in the circuit

E = 12 - 10 = 2 V

Total effective resistance,

r = 1 + 1 = 2 ohm

By using Ohm's law

E = I × R

I = 2 / 2 = 1 A

To find the current in the circuit, use Ohm's Law and solve two equations.

The current in the circuit can be found using Ohm's Law: V = IR, where V is the voltage, I is the current, and R is the resistance.

In this case, the emf of the first battery is 12 V with an internal resistance of 1 Ω. Let's assume the current in the circuit is I1. So, the terminal voltage across the first battery can be calculated using Ohm's Law: 12 = I1 * (1 + 1).

Similarly, for the second battery with emf of 10 V and internal resistance of 1 Ω, the terminal voltage can be calculated as 10 = (I1 - I) * (1 + 1), where I is the current flowing in the second battery.

Now, we need to solve these two equations to find the value of I, which is the current in the circuit.

Answer:

velocity maximum = 21.6 ft /s

frequency = 16.88 Hz

Explanation:

Amplitude, A = 0.24 in = 0.02 ft

maximum acceleration, a = 225 ft/s\

The formula for maximum acceleration is

a = ω² A

225 = ω² x 0.02

ω² = 11250

ω = 106.06 rad/s

Maximum velocity, v = ω A

v = 106.06 x 0.02 = 2.1 ft/s

Let f be the frequency

ω = 2 x 3.14 x f

f = 106.06 / (2 x 3.14) = 16.88 Hz

velocity maximum = 21.6 ft /s

frequency = 16.88 Hz

Amplitude, A = 0.24 in = 0.02 ft

maximum acceleration, a = 225 ft/s

The formula for maximum acceleration is

a = ω² A

225 = ω² x 0.02

ω² = 11250

ω = 106.06 rad/s

Maximum velocity, v = ω A

v = 106.06 x 0.02 = 2.1 ft/s

Let f be the frequency

ω = 2 x 3.14 x f

f = 106.06 / (2 x 3.14) = 16.88 Hz