A fan blade is rotating with a constant angular acceleration of +12.8 rad/s2. At what point on the blade, as measured from the axis of rotation, does the magnitude of the tangential acceleration equal that of the acceleration due to gravity? (g = 9.80 m/s2)

Solution:

To calculate the average distance between the given parabola and the x-axis

y = 77x(1616 - x)

x ∈ [0, 16]

avg distance = [tex]\int_{0}^{16}\frac{77x(1616 - x)dx}{\int_{0}^{16} x dx}[/tex]

                     = [tex]2\int_{0}^{16}(\frac{(124432x - 77x^{2})dx}{[x^{2}]_{0}^{16}}[/tex]

                      =[tex] 2\int_{0}^{16}\frac{\frac{124432x^{2}}{2}- \frac{77x^{3}}{3}}{[x^{2}]_{0}^{16}}[/tex]

                      = [tex]16^{2}[\frac{62216 - 25.67\times 16}{16^{2}}][/tex]

avg distance = 61805 unit

Answer:

The acceleration of each of the two blocks and the tension in the rope are 5.92 m/s and 147.44 N.

Explanation:

Given that,

Mass = 12.0 kg

Angle = 31.0°

Friction coefficient = 0.158

Mass of second block = 38.0 kg

Using formula of frictional force

[tex]f_{\mu} = \mu N[/tex]....(I)

Where, N = normal force

[tex]N = mg\cos\theta[/tex]

Put the value of N into the formula

[tex]N =12\times9.8\times\cos 31^{\circ}[/tex]

[tex]N=100.80\ N[/tex]

Put the value of N in equation (I)

[tex]f_{mu}=0.158\times100.80[/tex]

[tex]f_{mu}=15.9264\ N[/tex]

Now, Weight of second block

[tex]W = mg[/tex]

[tex]W=38.0\times9.8[/tex]

[tex]W=372.4\ N[/tex]

The horizontal force is

[tex]F = mg\sintheta[/tex]

[tex]F=12\times9.8\times\sin 31^{\circ}[/tex]

[tex]F=60.5684\ N[/tex]....(II)

(I). We need to calculate the acceleration

[tex]a=m_{2}g-\dfrac{f_{\mu}+mg\sin\theta}{m_{1}+m_{2}}[/tex]

[tex]a=\dfrac{372.4-(15.9264+60.5684)}{12+38}[/tex]

[tex]a=5.92\ m/s^2[/tex]

(II). We need to calculate the tension in the rope

[tex]m_{2}g-T=m_{2}a[/tex]

[tex]-T=38\times5.92-38\times9.8[/tex]

[tex]T=147.44\ N[/tex]

Hence, The acceleration of each of the two blocks and the tension in the rope are 5.92 m/s and 147.44 N.

Answer:

16.035 revolutions

Explanation:

Part 1:

t = 6 s, f0 = 0 , f = 5 rps,

Let the number of revolutions be n1.

Use first equation of motion for rotational motion

w = w0 + α t

2 x 3.14 x 5 = 0 + α x 6

α = 5.233 rad/s^2

Let θ1 be the angle turned.

Use second equation of motion for rotational motion

θ1 = w0 t + 12 x α x t^2

θ1 = 0 + 0.5 x 5.233 x 6 x 6 = 94.194 rad

n1 = θ1 / 2π = 94.194 / 2 x 3.14 = 15 revolutions

Part 2:

f0 = 5 rps, f = 0, t = 13 s

Let the number of revolutions be n2.

Use first equation of motion for rotational motion

w = w0 + α t

0 = 2 x 3.14 x 5 + α x 13

α = - 2.415 rad/s^2

Let θ2 be the angle turned.

Use third equation of motion for rotational motion

w^2 = w0^2 + 2 x α x θ2

0 = 2 x 3.14 x 5 - 2 x 2.415 x θ2

θ2 = 6.5 rad  

n2 = θ2 / 2π = 6.5 / 2 x 3.14 = 1.035 revolutions

total revolutions n = n1 + n2 = 15 + 1.035 = 16.035 revolutions

Answer:

Spring constant, k = 2.304 N/m

Explanation:

It is given that,

Mass of spring, m = 0.7 kg

Angular frequency, [tex]\omega=2.4\ rad/s[/tex]

We need to find the spring constant of the spring. It is a case of SHM. Its angular frequency is given by :

[tex]\omega=\sqrt{\dfrac{k}{m}}[/tex]

[tex]k=\omega^2\times m[/tex]

[tex]k=(2.4)^2\times 0.4\ kg[/tex]

k = 2.304 N/m

So, the spring constant of the spring is 2.304 N/m. Hence, this is the required solution.

Answer:

0.695 m

Explanation:

α = angular acceleration of the fan blade = 14.1 rad/s²

a = tangential acceleration at the point concerned = acceleration due to gravity = 9.8 m/s²

r = distance of the point from axis of rotation at which tangential acceleration is same as acceleration due to gravity

We know the relation between

a = r α

Inserting the values

9.8 = 14.1 r

r = 0.695 m

Answer:

[tex]a = 0.53 m/s^2[/tex]

Explanation:

initially the merry go round is at rest

after 6.73 s the merry go round will accelerates to 20 rpm

so final angular speed is given as

[tex]\omega = 2\pi f[/tex]

[tex]\omega = 2\pi ( \frac{20}{60})[/tex]

[tex]\omega = 2.10 rad/s[/tex]

so final tangential speed is given as

[tex]v = r\omega[/tex]

[tex]v = 1.71 (2.10) = 3.58 m/s[/tex]

now average acceleration of the girl is given as

[tex]a = \frac{v_f - v_i}{\Delta t}[/tex]

[tex]a = \frac{3.58 - 0}{6.73}[/tex]

[tex]a = 0.53 m/s^2[/tex]

Answer:

Beats are the difference in frequency.

(a) is correct option.

Explanation:

Beat :

Beat is the difference of the frequency of two waves.

The difference in frequency is equal to the number of beat per second.

Amplitude :

Amplitude of the wave is the maximum displacement.

Frequency :

Frequency is the number oscillations of wave in per second.

Intensity :

Intensity is the power per unit area.

Hence, Beats are the difference in frequency.

Answer:

It takes to a spacecraft 100,837.13 years to cross the galaxy if could travels at 99% the speed of light.

Explanation:

d= 9.461 *10²⁰m

V= 297 *10⁶ m/s

t= d/V

t= 3.18 * 10¹² seconds = 100837.13 years

Answer:

The velocity of the 4.00 kg object after the collision is 12 m/s.

Explanation:

Given that,

Mass of object [tex]m_{1} = 4.00\ kg[/tex]

Velocity of object [tex]v_{1} = 20.0\ m/s[/tex]

Mass of another object [tex]m_{2} = 6\ kg[/tex]

Velocity of another object [tex]v_{2}= 12.0\ m/s[/tex]

We need to calculate the relative velocity

[tex]v_{r}=v_{1}-v_{2}[/tex]

[tex]v_{r}=20-12=8\ m/s[/tex]

The relative velocity is 8 m/s in west before collision.

We know that,

In one dimensional elastic collision, the relative velocity before collision equals after collision but with opposite sign.

So, The relative velocity after collision must be 8 m/s in east.

So, The object of 6.00 kg is going 20 m/s and the object of 4.00 kg is slows down to 12 m/s.

Hence, The velocity of the 4.00 kg object after the collision is 12 m/s.

The 4.00-kg object is moving east at 2.00 m/s after the perfectly elastic collision with the 6.00-kg object.

This question is tackled using the concept of Conservation of Momentum. Since the collision is perfectly elastic, both the momentum and kinetic energy of the system are conserved.

In the east-west direction, momentum before is equal to momentum after the collision. Let's denote the velocity of the 4.00-kg object after the collision as v1. Thus, P(before) = P(after) gives us:
4.00kg * 20.00m/s (west) + 6.00kg * 12.00m/s (east) = 4.00kg * v1 (west) + 6.00kg * -12.00m/s (west).

Solving the above equation we find that v1 = -2.00 m/s, where the negative sign indicates it’s moving towards the east. Therefore, the 4.00-kg object is moving east at 2.00 m/s after the collision.

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The speed at which the satellite travels in its circular orbit around the Earth is 3,000 m/s.

In order to find the speed at which the satellite travels, we can use the formula:

v = 2πr / T

where v is the velocity of the satellite, r is the radius of the orbit, and T is the period of the satellite. Substituting the given values into the formula, we have:

v = (2π × 6,370,000) / (1.26 × 10^4) = 3,000 m/s

Therefore, the speed at which the satellite travels is 3,000 m/s.

The net work done by friction on the body of a snake slithering in a complete circle can be found by calculating the work done by the friction force. The friction force is equal to the coefficient of friction multiplied by the normal force. The net work done by friction is -2πr × friction force = -2π(1.04 m) × 19.5 N = -129.77 J.

The net work done by friction on the body of a snake slithering in a complete circle can be found by calculating the work done by the friction force. The friction force is equal to the coefficient of friction multiplied by the normal force. In this case, the friction force is opposing the motion of the snake, so the work done by friction is negative.

Considering a complete circle of radius 1.04 m, the distance traveled by the snake is equal to the circumference of the circle, which is 2πr. The normal force is equal to the weight of the snake, which is 78.0 N. The friction force is equal to the coefficient of friction (0.25) multiplied by the normal force (78.0 N), so the friction force is 19.5 N.

The net work done by friction is then calculated by multiplying the friction force by the distance traveled and taking into account the negative sign to indicate the opposing direction of friction. Therefore, the net work done by friction on the body of the snake is -2πr × friction force = -2π(1.04 m) × 19.5 N = -129.77 J.

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Answer:

The pressure of the air is 499.91 kPa.

Explanation:

Given that,

Initial pressure = 430 kPa

Temperature = 13.0+273=286 K

Final temperature = 59.5+273=332.5 K

We need to calculate the final pressure

Using relation of pressure and temperature

At constant volume,

[tex]\dfrac{P'}{P}=\dfrac{T'}{T}[/tex]

[tex]\dfrac{P'}{430}=\dfrac{332.5}{286}[/tex]

[tex]P'=\dfrac{332.5}{286}\times430[/tex]

[tex]P'=499.91\ kPa[/tex]

Hence,The pressure of the air is 499.91 kPa.

Answer:

W = 86 J

Explanation:

Work done by friction force is given by the formula

[tex]W = F_f . d[/tex]

here since the wagon is pulled by horizontal force such that speed remains constant

so here we will have

[tex]F_f = F_{ext} = 22 N[/tex]

Now we will have

[tex]W = 22\times 3.9 [/tex]

[tex]W = 86 J[/tex]

so work done against friction force will be equal to 86 J

The correct answer is option c 86.

Calculating Work Done by the Child:

In this  problem, we need to determine the amount of work done by a child who is pulling a 12 kg wagon over a distance of 3.9 meters with a force of 22 Newtons, assuming there is no friction.

The formula to calculate work (W) is:

W = F * d

where W is work, F is the force applied, and d is the distance covered.

Let's plug in the values provided:

F = 22 N
d = 3.9 m

Thus, the work done is:

W = 22 N * 3.9 m

W = 85.8 J

Therefore, the correct answer is approximately 86 J.

Answer:

68.445 cm³/s

Explanation:

Given:

Volume, V = [tex]\frac{4}{3}\pi r^3[/tex]

radius = 5.85 cm

Growth rate of radius = 0.5 cm/week

now

differentiating the volume with respect to time 't', we get:

[tex]\frac{dV}{dt}=\frac{d(\frac{4}{3}\pi r^3)}{dt}[/tex]

or

[tex]\frac{dV}{dt}=(\frac{4}{3}\pi )3r^2\frac{dr}{dt}[/tex]

now, substituting the value of r (i.e at r = 5.85cm) in the above equation, we get:

[tex]\frac{dV}{dt}=4\pi 5.85^2\times 0.5[/tex]

or

[tex]\frac{dV}{dt}=68.445cm^3/s[/tex]

hence, the rate of change of volume at r = 5.85cm is 68.445 cm³/s

Answer:

c

Explanation:

When a satellite is orbiting the earth , a constant force is being applied on it which means it must has acceleration. Also the direction of satellite is always being changed when it is orbitting to there is always change in the velocity vector which means acceleration.

You can view in the attached diagram to understand how the velocity is being changed.

Answers:

(a) [tex]0.0073kg[/tex]

(b) Volume gold: [tex]3.79(10)^{-7}m^{3}[/tex], Volume cupper: [tex]7.6(10)^{-8}m^{3}[/tex]

(c) [tex]17633.554kg/m^{3}[/tex]

Explanation:

We are told the total mass [tex]M[/tex] of the coin, which is an alloy  of gold and copper is:

[tex]M=m_{gold}+m_{copper}=7.988g=0.007988kg[/tex]   (1)

Where  [tex]m_{gold}[/tex] is the mass of gold and [tex]m_{copper}[/tex] is the mass of copper.

In addition we know it is a 22-karat gold and the relation between the number of karats [tex]K[/tex] and mass is:

[tex]K=24\frac{m_{gold}}{M}[/tex]   (2)

Finding [tex]{m_{gold}[/tex]:

[tex]m_{gold}=\frac{22}{24}M[/tex]   (3)

[tex]m_{gold}=\frac{22}{24}(0.007988kg)[/tex]   (4)

[tex]m_{gold}=0.0073kg[/tex]   (5)  This is the mass of gold in the coin

The density [tex]\rho[/tex] of an object is given by:

[tex]\rho=\frac{mass}{volume}[/tex]

If we want to find the volume, this expression changes to: [tex]volume=\frac{mass}{\rho}[/tex]

For gold, its volume [tex]V_{gold}[/tex] will be a relation between its mass [tex]m_{gold}[/tex]  (found in (5)) and its density [tex]\rho_{gold}=19.30g/cm^{3}=19300kg/m^{3}[/tex]:

[tex]V_{gold}=\frac{m_{gold}}{\rho_{gold}}[/tex]   (6)

[tex]V_{gold}=\frac{0.0073kg}{19300kg/m^{3}}[/tex]   (7)

[tex]V_{gold}=3.79(10)^{-7}m^{3}[/tex]   (8)  Volume of gold in the coin

For copper, its volume [tex]V_{copper}[/tex] will be a relation between its mass [tex]m_{copper}[/tex]  and its density [tex]\rho_{copper}=8.96g/cm^{3}=8960kg/m^{3}[/tex]:

[tex]V_{copper}=\frac{m_{copper}}{\rho_{copper}}[/tex]   (9)

The mass of copper can be found by isolating [tex]m_{copper}[/tex] from (1):

[tex]M=m_{gold}+m_{copper}[/tex]  

[tex]m_{copper}=M-m_{gold}[/tex]  (10)

Knowing the mass of gold found in (5):

[tex]m_{copper}=0.007988kg-0.0073kg=0.000688kg[/tex]  (11)

Now we can find the volume of copper:

[tex]V_{copper}=\frac{0.000688kg}{8960kg/m^{3}}[/tex]   (12)

[tex]V_{copper}=7.6(10)^{-8}m^{3}[/tex]   (13)  Volume of copper in the coin

Remembering density is a relation between mass and volume, in the case of the coin the density [tex]\rho_{coin[/tex] will be a relation between its total mass [tex]M[/tex] and its total volume [tex]V[/tex]:

[tex]\rho_{coin}=\frac{M}{V}[/tex] (14)

Knowing the total volume of the coin is:

[tex]V=V_{gold}+V_{copper}=3.79(10)^{-7}m^{3}+7.6(10)^{-8}m^{3}=4.53(10)^{-7}m^{3}[/tex] (15)

[tex]\rho_{coin}=\frac{0.007988kg}{4.53(10)^{-7}m^{3}}[/tex] (16)

Finally:

[tex]\rho_{coin}=17633.554kg/m^{3}}[/tex] (17)  This is the total density of the British sovereign coin

Answer:

[tex]B_2 = 1.71 mT[/tex]

Explanation:

As we know that the magnetic field near the center of solenoid is given as

[tex]B = \frac{\mu_0 N i}{L}[/tex]

now we know that initially the length of the solenoid is L = 18 cm and N number of turns are wounded on it

So the magnetic field at the center of the solenoid is 2 mT

now we pulled the coils apart and the length of solenoid is increased as L = 21 cm

so we have

[tex]\frac{B_1}{B_2} = \frac{L_2}{L_1}[/tex]

now plug in all values in it

[tex]\frac{2.0 mT}{B_2} = \frac{21}{18}[/tex]

[tex]B_2 = 1.71 mT[/tex]

The magnetic field strength inside a solenoid is dependent on the density of the coils per unit length. If a solenoid is extended, spreading the same number of coils over a greater distance, the magnetic field strength at its center decreases proportionally.

The magnetic field strength (B) inside a solenoid is proportional to the number of coils per unit length (n) and the current flowing through the coils (I). We can express this mathematically as B = μonI. The key concept here is that the field strength, B, depends on the density of the coils (n); in other words, how tightly or loosely coiled the wires are.

When you initially have a solenoid 18 cm long, and you extend it to 21 cm without changing the current or the size of the coils, the coil density per unit length decreases. Therefore, the magnetic field strength should decrease proportionally. In effect, you are spreading the same number of coils (field lines) over a greater distance, which dilutes the strength of the magnetic field at the solenoid's center. The exact strength would require knowing more specifics about the solenoid, but this discussion gives the general principle.

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Answer:

(a) 3.82 x 10⁷ m/s

(b) 4.5 MV/m

Explanation:

(a)

ΔV = change in the electric potential as the proton moves = 7.60 x 10⁶ Volts

q = magnitude of charge on proton = 1.6 x 10⁻¹⁹ C

v = speed gained by the proton

m = mass of proton = 1.67 x 10⁻²⁷ kg

Using conservation of energy

Kinetic energy gained by proton = Electric potential energy

(0.5) m v² = q ΔV

inserting the values

(0.5) (1.67 x 10⁻²⁷) v² = (1.6 x 10⁻¹⁹) (7.60 x 10⁶)

v = 3.82 x 10⁷ m/s

(b)

d = distance over which the potential change = 1.70 m

Electric field is given as

E = ΔV/d

E = 7.60 x 10⁶/1.70

E = 4.5 x 10⁶ V/m

E = 4.5 MV/m

Answer:

Momentum, p = 5 kg-m/s

Explanation:

The magnitude of the momentum of an object is the product of its mass m and speed v i.e.

p = m v

Mass, m = 3 kg

Velocity, v = 1.5 m/s

So, momentum of this object is given by :

[tex]p=3\ kg\times 1.5\ m/s[/tex]

p = 4.5 kg-m/s

or

p = 5 kg-m/s

So, the magnitude of momentum is 5 kg-m/s. Hence, this is the required solution.

Answer:

Final velocity is 181.61 m/s at angle 40.44° below horizontal.

Explanation:

Initial horizontal velocity = 170 cos 35.6 = 138.23 m/s

Final horizontal velocity = 138.23 m/s

Considering vertical motion of projectile:

Initial vertical velocity, u = 170 sin 35.6 = 98.96 m/s

Acceleration, a = -9.81 m/s²

Displacement, s = -208 m

We have v² = u² + 2as

Substituting

             v² = 98.96² + 2 x -9.81 x -208

             v = 117.79 m/s

Final velocity,

            [tex]v=\sqrt{138.23^2+117.79^2}=181.61m/s[/tex]

            [tex]\theta =tan^{-1}\left ( \frac{117.79}{138.23}\right )=40.44^0[/tex]

Final velocity is 181.61 m/s at angle 40.44° below horizontal.

Answer:

The distance is 5 m.

Explanation:

Given that,

Horizontal velocity = 20.0 m/s

Time t = 0.250 s

We need to calculate the horizontal distance

The distance traveled by the baseball with horizontal velocity in time t is given by

Using formula for horizontal distance

[tex]d = v\times t[/tex]

[tex]d = 20.0\times0.250[/tex]

[tex]d=5\ m[/tex]

Hence, The distance is 5 m.

Answer:

23.4 m/s

Explanation:

f = actual frequency of the wave = 6.2 x 10⁹ Hz

[tex]f_{app}[/tex] = frequency observed as the ball approach the radar

[tex]f_{rec}[/tex] = frequency observed as the ball recede away from the radar

V = speed of light

[tex]v[/tex] = speed of ball

B = beat frequency = 969 Hz

frequency observed as the ball approach the radar is given as

[tex]f_{app}=\frac{f(V+v)}{V}[/tex]                                 eq-1

frequency observed as the ball recede the radar is given as

[tex]f_{rec}=\frac{f(V-v)}{V}[/tex]                                  eq-2

Beat frequency is given as

[tex]B = f_{app} - f_{rec}[/tex]

Using eq-2 and eq-1

[tex]B = \frac{f(V+v)}{V}- \frac{f(V-v)}{V}[/tex]

inserting the values

[tex]969 = \frac{(6.2\times 10^{9})((3\times 10^{8})+v)}{(3\times 10^{8})}- \frac{(6.2\times 10^{9})((3\times 10^{8})-v)}{(3\times 10^{8})}[/tex]

[tex]v[/tex] = 23.4 m/s

Answer:

option d)

Explanation:

The amount of heat required to convert the 1 g of ice at 0 degree C to 1 g water at 0 degree C is called latent heta of fusion.

H = m Lf

where,, h is the heat required, m is the mass and Lf is the latent heat of fusion

1140 = m x 79.7

m = 14.3 g

Answer:

150 fils

Explanation:

Power = 125 Watt

Time 4 hours a day

Energy per day = Power x time per day = 125 x 4 = 500 watt hour

Energy for 30 days = 500 x 30 watt hour = 15000 watt hour = 15 KWH

Cost of 1 KWH = 10 fils

Cost of 15 KWH = 15 x 10 = 150 fils

It takes approximately 2.80 seconds for the second-place cyclist to catch up with the leader.

To find the time it takes for the second-place cyclist to catch up with the leader, we can set up an equation based on their positions.

Let's assume that the time it takes for the second-place cyclist to catch up is "t" seconds.

For the leader:

Distance = Velocity × Time

Distance(leader) = Velocity(leader) × t

For the second-place cyclist:

Distance = Initial Distance + Velocity × Time + 0.5 × Acceleration × Time²

Distance(second) = Initial Distance(second) + Velocity(second) × t + 0.5 × Acceleration(second) × t²

Given:

Velocity(leader) = 11.9 m/s

Initial Distance(second) = 10.6 m

Velocity(second) = 9.80 m/s

Acceleration(second) = 1.20 m/s²

Equating the two distances:

Distance(leader) = Distance(second)

Velocity(leader) × t = Initial Distance(second) + Velocity(second) × t + 0.5 × Acceleration(second) × t²

Now, plug in the values and solve for "t":

11.9t = 10.6 + 9.80t + 0.5 × 1.20 × t²

Simplify the equation:

0.6t² + 2.1t - 10.6 = 0

solving the equation we get,

t = 2.80292945

t = -6.2898

Since time can not be negative so considering t = 2.80292945.

So, it takes approximately 2.80 seconds for the second-place cyclist to catch up with the leader.

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To determine the time elapsed before the second-place cyclist catches the leader in a bicycle race, we use the equations of motion to calculate when their positions will be equal, accounting for the leader's constant velocity and the second-place cyclist's acceleration.

The problem at hand involves the second-place cyclist catching up to the leader in a bicycle race. Given that the leader has a constant velocity of 11.9 m/s and the gap between them is 10.6 m, while the second-place cyclist has a velocity of 9.80 m/s and an acceleration of 1.20 m/s2, we can find out the time it takes for the second-place cyclist to catch up by using the equations of motion.

The equation for the leader's position as a function of time will be:
x1(t) = 11.9t + 10.6 ,
and for the second-place cyclist, considering initial velocity (u), acceleration (a), and the initial position being 0, the position as a function of time is:
x2(t) = 0 + 9.80t + 0.5(1.20)t2.

To find the time when the second-place cyclist catches up to the leader, we need to set x1(t) = x2(t) and solve for t:

11.9t + 10.6 = 9.80t + 0.5(1.20)t2

Solving this quadratic equation will give us the time elapsed before the second-place cyclist catches the leader.

To change the wave speed on a guitar string from 328 m/s to 363 m/s, the tension must be increased by 22.04%, as calculated through the wave speed formula for a string, taking into account the ratio of the squares of the wave speeds.

To find by what percent the tension in a guitar string must increase to change the wave speed on the string from 328 m/s to 363 m/s, we can use the formula for wave speed on a string, which is [tex]v = (\sqrt{T/mu}\)[/tex], where v is the wave speed in meters per second, T is the tension in Newtons, and [tex](mu)[/tex] is the linear mass density in kilograms per meter.

Starting with the initial tension (let's call it T1) and the initial wave speed (328 m/s), we have:

[tex]v1 = \(\sqrt{T1/\mu}\)[/tex]

For the final wave speed (363 m/s), we have:

[tex]v2 = \(\sqrt{T2/mu}\)[/tex]

Using the ratio of the squares of the wave speeds:

[tex]\((v2/v1)^2 = (T2/T1)[/tex]

[tex]\((363/328)^2 = (T2/T1)[/tex]

[tex]\(T2/T1 = 1.2204[/tex]

To find the percentage increase in tension:

Percent increase = [tex]\((T2/T1 - 1) \times 100\%)[/tex]

Percent increase = [tex]\((1.2204 - 1) \times 100\%\)[/tex]

Percent increase = 22.04%

Therefore, the tension in the guitar string must be increased by 22.04% to achieve a wave speed change from 328 m/s to 363 m/s.

Answer:

(a) 7.92 A

(b) 0.133 Nm

Explanation:

N = 216

r = 2.32 cm = 0.0232 m

(a) M = 2.89 Am^2

M = N i A

Where, A be the area of the coil and i be the current in the coil

2.89 = 216 x i x 3.14 x 0.0232 x 0.0232

i = 7.92 A

(b) B = 46 mT = 0.046 T

Torque, τ = M B Sin 90

τ = 2.89 x 0.046 x 1 = 0.133 Nm

The current required to produce a magnetic dipole moment of 2.89 A·m² in a coil of 216 turns and radius 2.32 cm is approximately 7.92 A. The maximum torque this coil can experience in a 46.0 mT magnetic field is about 0.133 N·m.

To solve these problems, we'll use the relationship between the magnetic dipole moment, current, and torque in a magnetic field.

Part (a): Calculate the Current

The magnetic dipole moment (μ) of a coil is given by:

μ = NIA

where N is the number of turns, I is the current, and A is the area of the coil.

Given:

N = 216 turns

μ = 2.89 A·m²

Radius (r) = 2.32 cm = 0.0232 m

The area (A) of the coil is:

A = πr² = π(0.0232 m)² ≈ 1.69 × 10⁻³ m²

To find the current (I), rearrange the formula:

I = μ / (N × A)

Substitute the values:

I = 2.89 A·m² / (216 × 1.69 × 10⁻³ m²) ≈ 7.92 A

Part (b): Calculate the Maximum Torque

The maximum torque (τ) experienced by the coil in a magnetic field (B) is given by:

τ = μB

Given the magnetic field (B) is 46.0 mT = 0.046 T, substitute the values:

τ = 2.89 A·m² × 0.046 T ≈ 0.133 N·m

Thus, the current required to achieve the given magnetic dipole moment is approximately 7.92 A, and the maximum torque experienced by the coil in a 46.0 mT magnetic field is approximately 0.133 N·m.

Answer:

(a) 328 Nm

(b) 79.35 Nm

Explanation:

N = =150, side = 17.5 cm = 0.175 m, i = 42 A, B = 1.7 T

A = side^2 = 0.175^2 = 0.030625 m^2

(a) Torque = N x i x A x B x Sinθ

For maximum torque, θ = 90 degree

Torque = 150 x 42 x 0.030625 x 1.7 x Sin 90

Torque = 328 Nm

(b) θ = 14 degree

Torque =  150 x 42 x 0.030625 x 1.7 x Sin 14

Torque = 79.35 Nm

The torque on a current-carrying loop in a magnetic field is calculated using the formula T = NIAB sin ϴ. Part A finds the maximum torque by setting ϴ to 90 degrees, and Part B calculates the torque at an angle of 14 degrees by applying the sine of that angle.

To calculate the torque on a current-carrying loop in a magnetic field, we use the formula T = NIAB sin ϴ, where T is the torque, N is the number of turns in the coil, I is the current, A is the area, B is the magnetic field strength, and ϴ is the angle between the magnetic field and the normal to the plane of the loop.

Part A: Maximum Torque

To find the maximum torque, we set ϴ to 90°, as sin(90°) = 1. Thus, Tmax = NIAB. Using the given values: N = 150 turns, I = 42 A, A (area of the square loop) = (0.175 m)2 and B = 1.7 T, we calculate the maximum torque.

Part B: Torque at 14°

To find the torque when ϴ is 14°, sin(14°) is used in the formula. We thus get T = NIAB sin(14°).

1. Area of Circular End: [tex]A = π x (1.5)^2 = 7.07 sq ft.[/tex]

2. Weight of Gasoline: W = 133.5π lbs.

3. Fluid Force: F = 2043.45π lbs on half-full tank end.

let's break down the problem step by step.

1. Identify Relevant Parameters:

  - Diameter of the tank (D) = 3 feet

  - Radius of the tank (r) = D/2 = 1.5 feet

  - Density of gasoline (ρ) = 42 pounds per cubic foot

  - Depth of the gasoline (h) when the tank is half full = 1.5 feet

2. Calculate the Area of the Circular End:

  - The area (A) of a circle is given by the formula:[tex]A = π x r^2[/tex]

  - Substituting the value of radius (r = 1.5 feet), we get:

  - [tex]A = πx (1.5)^2 = π x 2.25 = 7.07[/tex] square feet

3. Determine the Weight of the Gasoline:

  - The weight (W) of the gasoline can be calculated using the formula: W = ρx V

  - Where V is the volume of the gasoline.

  - Since the tank is half full, the volume of the gasoline is half the volume of the cylinder.

  - The volume (V) of the cylinder is given by the formula: [tex]V = πxr^2 x h[/tex]

  - Substituting the values of radius (r = 1.5 feet) and height (h = 1.5 feet), we get:

[tex]- V = π x (1.5)^2 x 1.5 = π x 2.25x 1.5 = 3.375π cubic feet[/tex]

  - So, the weight of the gasoline (W) is: W = 42x3.375π = 133.5π pounds

4. Calculate the Fluid Force:

  - The fluid force (F) on a circular end of the tank can be calculated using the formula: F = ρ x g x V x h

  - Where g is the acceleration due to gravity.

  - Since we're working in pounds and feet, we'll use the value of acceleration due to gravity, [tex]g ≈ 32.2 ft/s^2.[/tex]

  - Substituting the values, we get:

  - F = 42x 32.2x3.375π x 1.5 = 2043.45π pounds

5.Final Calculation:

  - Therefore, the fluid force on a circular end of the tank when the tank is half full is approximately 2043.45π pounds.

So, the complete calculation and explanation show that the fluid force on a circular end of the tank when it is half full is about 2043.45π pounds.

Answer:

5.3 x 10⁻⁹ C

Explanation:

r = radius of cylindrical shell = 10⁻⁵ m

L = length = 0.32 m

A = area

Area is given as

A = 2πrL

A = 2 (3.14) (10⁻⁵) (0.32)

A = 20.096 x 10⁻⁶ m²

d = separation = 10⁻⁸ m

[tex] k_{appa} [/tex] = dielectric constant = 4

Capacitance is given as

[tex]Q=\frac{k_{appa}\epsilon _{o}A}{d}[/tex]                               eq-1

V = Potential difference across the membrane = 74 mV = 0.074 Volts

Q = magnitude of charge on each side

Magnitude of charge on each side is given as

Q = CV

using eq-1

[tex]Q=\frac{k_{appa} \epsilon _{o}AV}{d}[/tex]

Inserting the values

[tex]Q=\frac{4 (8.85\times 10^{-12})(20.096\times 10^{-6})(0.074)}{10^{-8}}[/tex]

Q = 5.3 x 10⁻⁹ C

The membrane is parallel and act as a parallel plate capicitor.The magnitude of the charge on each side of the membrane will be 5.3×10⁻⁹C.

It is an type capacitor is an in which two metal plates arranged in such away so that they are connected in parallel andhaving some distance between them.

A dielectric medium is must in between these plates help to stop the flow of electric current through it due to its non-conductive nature .

The given data in the problem is;

r is the radius of cell of thin cyliner=10⁻⁵m

L is the length =0.32

d is the thickness =10⁻⁸m

K is the dielectric constant=4

v is the potential difference across the membrane=74 mv=0.074 v

q is the megnitude of charge=?

A is the area of capicitor=2πrl

[tex]\rm A = 2\pi rl\\\\\rm A= 2\times3.14\times10^{-5}\times0.32\\\\\rm A= 20.096\times10^{-6}[/tex]

The given formula for the parallel plate capicitor as ,

[tex]\rm Q=\frac{K\varepsilon _0AEr}{d} \\\\\rm Q=\frac{4\times(8.85\times10^{-12})(20.096\times10^{-6}\times0.074}{10^{-8}}\\\\\rm Q=5.3\TIMES10^{-9}\;C[/tex]

Hence the charge on each side of the membrane will be 5.3×10⁻⁹C.

To learn more about the parallel plate capicitor refer to the link;

https://brainly.com/question/12883102