Heated lithium atoms emit photons of light with an energy of 2.961 × 10−19 J. Calculate the frequency and wavelength of one of these photons. What is the total energy in 1 mole of these photons? What is the color of the emitted light?

Answer:

Coefficient of static friction between the tires and the road is 0.92.

Explanation:

It is given that,

Radius of the curve, r = 120 m

Speed, v = 119 km/h = 33.05 m/s

We need to find the coefficient of static friction between the tires and the road. In a curved road the safe velocity is given by :

[tex]v=\sqrt{\mu rg}[/tex]

[tex]\mu[/tex] is the coefficient of static friction

g is acceleration due to gravity

[tex]\mu=\dfrac{v^2}{rg}[/tex]

[tex]\mu=\dfrac{(33.05\ m/s)^2}{120\ m\times 9.8\ m/s^2}[/tex]

[tex]\mu=0.92[/tex]

So, the coefficient of static friction between the tires and the road is 0.92. Hence, this is the required solution.

Answer:

B = 0.75 T

Explanation:

As we know that the force on a moving charge in magnetic field is given by the formula

[tex]F = qvB[/tex]

here we have

[tex]B = \frac{F}{qv}[/tex]

here we know that

[tex]F = 4.8 \times 10^{-13} N[/tex]

[tex]q = 1.6 \times 10^{-19} C[/tex]

[tex]v = 4 \times 10^6 m/s[/tex]

now from above equation we have

[tex]B = \frac{4.8 \times 10^{-13}}{(1.6 \times 10^{-19})(4 \times 10^6)}[/tex]

[tex]B = 0.75 T[/tex]

Answer:

 D . 1.47 x 10⁻² Ω-m

Explanation:

L = length of the cylindrical resistor = 2 m

d = diameter = 0.1 m

A = Area of cross-section of the resistor = (0.25) [tex]\pi[/tex] d² = (0.25) (3.14) (0.1)² = 0.785 x 10⁻² m²

V = battery Voltage = 12 volts

[tex]i [/tex] = current flowing through the resistor = 3.2 A

R = resistance of the resistor

Resistance of the resistor is given as

[tex]R = \frac{V}{i}[/tex]

[tex]R = \frac{12}{3.2}[/tex]

R = 3.75 Ω

[tex]\rho[/tex] = resistivity

Resistance is also given as

[tex]R = \frac{ \rho L}{A}[/tex]

[tex]3.75 = \frac{ \rho (2)}{(0.785\times 10^{-2})}[/tex]

[tex]\rho[/tex] =  1.47 x 10⁻² Ω-m

Answer:

517.6 N

8.63 m/s²

Explanation:

M = mass of earth = 5.98 x 10²⁴ kg

m = mass of the person = 60 kg

W = weight of the person

R = radius of earth = 6.4 x 10⁶ m

d = distance of ISS above the surface of earth = 400 km = 4 x 10⁵ m

Weight of person on the ISS is given as

[tex]W = \frac{GMm}{(R+d)^{2}}[/tex]

[tex]W = \frac{(6.67\times 10^{-11})(5.98\times 10^{24})(60)}{((6.4\times 10^{6})+(4\times 10^{5}))^{2}}[/tex]

W = 517.6 N

Acceleration due to gravity is given as

[tex]g = \frac{GM}{(R+d)^{2}}[/tex]

[tex]g = \frac{(6.67\times 10^{-11})(5.98\times 10^{24})}{((6.4\times 10^{6})+(4\times 10^{5}))^{2}}[/tex]

g = 8.63 m/s²

Answer:

kinematic viscosity in SUS is = 671.64 SUS

Explanation:

given data

kinetic viscosity = 145 mm^2/s

we know

1 mm = 0.1 cm

so kinetic viscosity in cm is [tex]\nu =145 (0.1)^{2} =1.45 cm^{2}/s[/tex]

other unit of kinetic viscosity is centistokes

[tex]1 cm^{2}/s = 100 cst[/tex]

so 1.45 cm^2/s will be 145 cst

if the temperature is 260°f , then cst value should be multiplied by 4.632. therefore kinematic viscosity in SUS is = 4.362 *145 = 671.64 SUS

Answer: [tex]22.8^0C[/tex]

Explanation:-

[tex]Q_{absorbed}=Q_{released}[/tex]

As we know that,  

[tex]Q=m\times c\times \Delta T=m\times c\times (T_{final}-T_{initial})[/tex]

[tex]m_1\times c\times (T_{final}-T_1)=-[m_2\times c\times (T_{final}-T_2)][/tex]      

where,

[tex]m_1[/tex] = mass of iron horseshoe = 0.35 kg = 350 g (1kg=1000g[/tex]

[tex]m_2[/tex] = mass of water = 21.9 kg = 21900 g

[tex]T_{final}[/tex] = final temperature = ?

[tex]T_1[/tex] = temperature of iron horseshoe = [tex]600^oC[/tex]

[tex]T_2[/tex] = temperature of water = [tex]21.8^oC[/tex]

[tex]c_1[/tex] = specific heat of iron horseshoe = [tex]0.450J/g^0C[/tex]

[tex]c_2[/tex] = specific heat of water =  [tex]4.184J/g^0C[/tex]

Now put all the given values in equation (1), we get

[tex]m_1\times c_1\times (T_{final}-T_1)=-[m_2\times c_2\times (T_{final}-T_2)][/tex]

[tex]350\times 0.450\times (T_{final}-600)^0C=-[21900g\times 4.184\times (T_{final}-21.8)][/tex]

[tex]T_{final}=22.8^0C[/tex]

Therefore, the final equilibrium temperature is [tex]22.8^0C[/tex].

Answer:

1.718 N , attractive

Explanation:

r = 0.66 m, n = 5.7 x 10^13

q1 = 5.7 x 10^13 x 1.6 x 10^-19 = 9.12 x 10^-6 C

q2 = - 5.7 x 10^13 x 1.6 x 10^-19 = - 9.12 x 10^-6 C

F = K q1 q2 / r^2

F = 9 x 10^9 x  9.12 x 10^-6 x 9.12 x 10^-6 / (0.66)^2

F = 1.718 N

As both the charges are opposite in nature, so the force between them is attractive.

Answer:

The time period of the wave is 0.002 sec.

Explanation:

Given that,

Frequency = 0.5 kHz

Phase difference [tex]\phi=\dfrac{\pi}{3}[/tex]

Path difference = 1.5 cm

We need to calculate the time period

Using formula of time period

The frequency is the reciprocal of time period.

[tex]f =\dfrac{1}{T}[/tex]

[tex]T=\dfrac{1}{f}[/tex]

Where, f = frequency

Put the value into the formula

[tex]T=\dfrac{1}{0.5\times10^{3}}[/tex]

[tex]T=0.002\ sec[/tex]

Hence, The time period of the wave is 0.002 sec.

Answer:

[tex]x_{1}=-2,x_{2}=4[/tex]

value of g(x) at these points are as follows

g(-2)=30

g(4)=-78

Explanation:

Given

g(x)=[tex]x^{3}-3x^{2}-24x+2[/tex]

Differentiating with respect to x we get

[tex]g'(x)=3x^{2}-6x-24[/tex]

to obtain point of extrema we equate g'(x) to zero

[tex]g'(x)=3x^{2}-6x-24\\\\\therefore 3x^{2}-6x-24=0\\\\\Rightarrow x^{2}-2x-8=0\\\\x^{2}-4x+2x-8=0\\x(x-4)+2(x-4)=0\\(x+2)(x-4)=0[/tex]

Thus the critical points are obtained as [tex]x_{1}=-2,x_{2}=4[/tex]

The values at these points are as

[tex]g(-2)=(-2)^{3}-3(-2)^{2}-24(-2)+2=30\\\\g(4)=(4)^{3}-3(4)^{2}-24(4)+2=-78[/tex]

Answer:

The difference in the length of the two rods when compressed is [tex]5.4\times10^{-5}\ m[/tex].

Explanation:

Given that,

Length = 0.780 m

Diameter = 1.50 cm

Force = 4350 N

(a). For steel rod

We know ,

The young modulus for steel rod

[tex]Y=2\times10^{11}[/tex]

Using formula of young modulus

[tex]e_{s}=\dfrac{Fl}{AY}[/tex]

[tex]e_{s}=\dfrac{4350\times0.780}{3.14\times(0.75\times10^{-2})^2\times2\times10^{11}}[/tex]

[tex]e_{s}=9.6\times10^{-5}\ m[/tex]

(b). For copper rod

We know ,

The young modulus for steel rod

[tex]Y=1.1\times10^{11}[/tex]

Using formula of young modulus

[tex]e_{c}=\dfrac{Fl}{AY}[/tex]

[tex]e_{c}=\dfrac{4350\times0.780}{3.14\times(0.75\times10^{-2})^2\times1.1\times10^{11}}[/tex]

[tex]e_{c}=1.5\times10^{-4}\ m[/tex]

The difference in the length of the two rods when compressed is

[tex]difference\ in\ length=e_{c}-e_{s}[/tex]

[tex]difference\ in\ length=1.5\times10^{-4}-9.6\times10^{-5}[/tex]

[tex]difference\ in\ length =5.4\times10^{-5}\ m[/tex]

Hence, The difference in the length of the two rods when compressed is [tex]5.4\times10^{-5}\ m[/tex].

Answer:

[tex]K = 0.076 W/m K[/tex]

Explanation:

Heat required to melt the complete ice is given as

[tex]Q = mL[/tex]

here we have

m = 5.0 kg

[tex]L = 3.35 \times 10^5 J/kg[/tex]

now we have

[tex]Q = (5 kg)(3.35 \times 10^5)[/tex]

[tex]Q = 1.67 \times 10^6 J[/tex]

now the power required to melt ice in 8.4 hours is

[tex]P = \frac{Q}{t} = \frac{1.67\times 10^6}{8.4 \times 3600 s}[/tex]

[tex]P = 55.4 Watt[/tex]

now by formula of conduction we know

[tex]P = \frac{KA(\Delta T)}{x}[/tex]

now we have

[tex]55.4 = \frac{K(0.73 m^2)(25 - 5)}{0.02}[/tex]

[tex]K = 0.076 W/m K[/tex]

Answer:

3.62 m  and - 1.4 m

Explanation:

Consider a location towards the positive side of x-axis beyond the location of charge Q₂

x = distance of the location from charge Q₂

d = distance between the two charges = 2 m

For the electric field to be zero at the location

E₁ = Electric field by charge Q₁ at the location = E₂ = Electric field by charge Q₂ at the location

[tex]\frac{kQ_{1}}{(2 + x)^{2}}= \frac{kQ_{2}}{x^{2}}[/tex]

[tex]\frac{2.5\times 10^{-5}}{(2 + x)^{2}}= \frac{5 \times 10^{-6}}{x^{2}}[/tex]

x = 1.62 m

So location is 2 + 1.62 = 3.62 m

Consider a location towards the negative side of x-axis beyond the location of charge Q₁

x = distance of the location from charge Q₁

d = distance between the two charges = 2 m

For the electric field to be zero at the location

E₁ = Electric field by charge Q₁ at the location = E₂ = Electric field by charge Q₂ at the location

[tex]\frac{kQ_{1}}{(x)^{2}}= \frac{kQ_{2}}{ (2 + x)^{2}}[/tex]

[tex]\frac{2.5\times 10^{-5}}{(x)^{2}}= \frac{5 \times 10^{-6}}{(2+x)^{2}}[/tex]

x = - 1.4 m

Electric field is zero due to positive point charge Q1 and negative Q2 along the x axis is at the location of 3.62 meters.

The electric field is the field, which is surrounded by the electric charged. The electric field is the electric force per unit charge.

Given information-

The charge of the point 1 is [tex]2.5\times10^{-5} \rm C[/tex].

The charge of the point 2 is [tex]-5.0\times10^{-6} \rm C[/tex].

The distance between the point 1 and point 2 is 2 meters away from the x axis.

Let the position of the point 1 is at x.

Thus the electric force on point Q1 is,

[tex]F_1=\dfrac{kQ1}{x^2}[/tex]

As the distance between the point Q1 and point Q2 is 2 meters away from the x axis. Thus the position of it should be at (x+2). The electric force on point 2

[tex]F_2=\dfrac{kQ1}{(x+2)^2}[/tex]

As the force of two is equal and opposite thus,

[tex]\dfrac{kQ1}{x^2}=\dfrac{kQ2}{(x+2)^2}\\\dfrac{2.5\times10^{-5}}{x^2}=\dfrac{-5\times10^{-6}}{(x+2)^2}\\x=1.62[/tex]

Thus the position of point 2 is,

[tex]p_2=2+1.62\\p_2=3.62\rm m[/tex]

Thus, the location of the place along the x axis where the electric field due to these two charges is zero is 3.62 meters.

Learn more about electric field here;

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Answer:

[tex]\mu = 1.39[/tex]

Explanation:

Since the reflection is nearly zero intensity

so here we will say that the reflected light must show destructive interference

so here we have

[tex]path \: difference = \frac{\lambda}{2}[/tex]

[tex]2 \mu t = \frac{\lambda}{2}[/tex]

[tex]t = \frac{\lambda}{4\mu}[/tex]

here we have

[tex]\lambda = 675 nm[/tex]

t = 121 nm

now from above equation we have

[tex]\mu = \frac{675 nm}{4(121 nm)}[/tex]

[tex]\mu = 1.39[/tex]

Answer:

1.59 m/s^2, 65.2°

Explanation:

F1 = 390 N North

F2 = 180 N east

m = 270 kg

Net force is the vector sum of both the forces.

[tex]F = \sqrt{F_{1}^{2}+F_{2}^{2}}[/tex]

[tex]F = \sqrt{390^{2}+180^{2}}[/tex]

F = 429.53 N

Direction of force

tan∅ = F1 / F2 = 390 / 180 = 2.1667

∅ = 65.2°

The direction of acceleration is same as the direction of net force.

The magnitude of acceleration is

a = F / m = 429.53 / 270 = 1.59 m/s^2

Final answer:

The magnitude of the sailboat's acceleration is 1.59 m/s², and the direction is 25 degrees east of north. This is calculated using Newton's second law and vector addition of the orthogonal forces exerted by the wind and water.

Explanation:

To calculate the magnitude and direction of the sailboat's acceleration, we need to use Newton's second law, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Here, we combine the forces exerted by the wind and the water to obtain the net force. The net force can be calculated using vector addition, where the force due to the wind (390 N north) and the force due to the water (180 N east) are treated as orthogonal vectors.

The net force (Fnet) is the vector sum of the two individual forces. We calculate the net force using the Pythagorean theorem:

Fnet = √Fwind² + Fwater² = √390² + 180² = √152100 + 32400 = √184500 N

So, the magnitude of the net force is approximately 429.53 N. To find the acceleration (a), we use the formula:

a = Fnet / m

Substituting the mass of the sailboat (270 kg) and the net force, we get:

a = 429.53 N / 270 kg = 1.59 m/s²

To determine the direction of the acceleration, we take the arctangent of the ratio of the forces. Since arctan(180/390) equals approximately 25 degrees, this is the angle east of north.

The sailboat's acceleration has a magnitude of 1.59 m/s² and a direction of 25 degrees east of north.

Using the kinematic equations of motion, we can calculate that a projectile launched at an initial speed of 1360 m/s from the moon's surface will reach two-fifths its initial speed at an altitude of roughly 680 kilometers.

A projectile launched vertically from the moon's surface decelerates due to the moon's gravity at a rate of 1.6 m/s². Given the initial speed of the projectile is 1360 m/s, we want to find out the altitude at which the speed is two-fifths of this initial velocity. To find out, we use the kinematic equation v² = u² + 2as.

Substituting the values into the equation, we get (2/5 * 1360)² = (1360)² - 2 * 1.6 * s. Solving this equation gives us the distance s, where s comes out to be approximately 680000 meters or 680 km.

This means that the projectile's speed is two-fifths its initial value at an altitude of approximately 680 km from the moon's surface.

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To find the altitude at which the projectile's speed is two-fifths its initial value, the loss of kinetic energy must equal the gain in potential energy. Solving for the altitude (h), we find the answer.

This problem involves physics concepts of projectile motion and gravitational acceleration. Given that the projectile is launched with an initial velocity of 1360 m/s, and we want to determine the altitude at which it has slowed to two-fifths of that speed, we will be considering the effects of the Moon's gravity on the deceleration of the projectile.

Using the formula for kinetic energy K.E = 1/2 m v^2, when the projectile's velocity slows to two-fifths of its original speed, the kinetic energy will be four-fifths less since kinetic energy is proportional to the square of the speed. This loss of kinetic energy must equal the gain in potential energy (since energy is conserved), which is given by the formula P.E = mgh (where m is mass, g is gravitational acceleration, and h is height or altitude).

Solving P.E = K.E for h, we get the altitude at which the projectile's speed is two-fifths its initial value.

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Answer:

Part a)

[tex]f = 1.46 \times 10^7 Hz[/tex]

Part b)

[tex]KE = 3.87 \times 10^{-12} J[/tex]

Explanation:

Part a)

As we know that radius of circular path of a charge moving in constant magnetic field is given as

[tex]R = \frac{mv}{qB}[/tex]

now we have

[tex]v = \frac{qBR}{m}[/tex]

now the frequency of oscillator is given as

[tex]f = \frac{v}{2\pi R}[/tex]

[tex]f = \frac{qB}{2\pi m}[/tex]

[tex]f = \frac{(1.6 \times 10^{-19})(0.960)}{2\pi(1.67\times 10^{-27})}[/tex]

[tex]f = 1.46 \times 10^7 Hz[/tex]

PART b)

now for kinetic energy of proton we will have

[tex]KE = \frac{1}{2}mv^2[/tex]

[tex]KE = \frac{1}{2}m(\frac{qBR}{m})^2[/tex]

[tex]KE = \frac{q^2B^2R^2}{2m}[/tex]

[tex]KE = \frac{(1.6 \times 10^{-19})^2(0.960)^2(0.740)^2}{2(1.67\times 10^{-27})}[/tex]

[tex]KE = 3.87 \times 10^{-12} J[/tex]

Answer:

The internal energy is 73.20 J.

Explanation:

Given that,

Weight of helium = 86 mg

Temperature = 0°C

We need to calculate the internal energy

Using formula of internal energy

[tex]U =nc_{v}T[/tex]

Where, [tex]c_{v}[/tex] = specific heat at constant volume

He is mono atomic.

So, The value of [tex]c_{v}= \dfrac{3}{2}R[/tex]

now, 1 mole of Helium = 4 g helium

n =number of mole of the 86 mg of helium

[tex]n = \dfrac{86\times10^{-3}}{4}[/tex]

[tex]n =2.15\times10^{-2}\ mole[/tex]

T = 0°C=273 K

Put the value into the formula

[tex]U = 2.15\times10^{-2}\times\dfrac{3}{2}\times8.314\times273[/tex]

[tex]U = 73.20\ J[/tex]

Hence, The internal energy is 73.20 J.

Answer:

3.91 N/C

Explanation:

u = 0, s = 5.50 m, t = 4 us = 4 x 10^-6 s

Let a be the acceleration.

Use second equation of motion

s = u t + 1/2 a t^2

5.5 = 0 + 1/2 a (4 x 10^-6)^2

a = 6.875 x 10^11 m/s^2

F = m a

The electrostatic force, Fe = q E

Where E be the strength of electric field.

So, q E = m a

E = m a / q

E = (9.1 x 10^-31 x 6.875 x 10^11) / ( 1.6 x 10^-19)

E = 3.91 N/C

Answer:

2.5 min

Explanation:

Hello

by definition  the power   is the amount of work done per unit of time. in this case, we know  the total power and the work developed

[tex]1 Watt= \frac{Joule }{sec}[/tex]

Let

[tex]P=1200 W\\E=1.8 *10^{5}\\X=\frac{1.8 *10^{5} j }{1200 W}\\X= unknown\ time\ (sec) \\\\we\ need\ the\ answer\ in\ minutes,\\ 1\ min=60\ sec\\\\x=150 sec \\150\ sec*(\frac{1 min}{ 60 sec})=2.5 min\\[/tex]

hence , the data is not altered, we did it like this to eliminate the sec units.

Answer 2.5 min

I hope it helps

The food was in the 1200 W microwave for 150 seconds, which is equivalent to 2.5 minutes.

To calculate how long the food was in the microwave, we use the formula for power, which is the rate at which work is done or energy is transferred:

Power (P) = Energy (E) / Time (t)

First, rearrange this formula to solve for time (t):

t = E / P

Plugging in the given values:

So:

t = (1.8 times 10⁵J) / (1200 W)

Calculate this to find the time in seconds, and then convert it to minutes as follows:

t = 150 seconds

Divide by 60 to get minutes:

t = 2.5 minutes

Answer:

True

Explanation:

The force on the electron when it enters in a magnetic field is given by

F = q ( v x B)

F = -e x V x B x Sin∅

here, F is the force vector, B be the magnetic field vector and v be the velocity vector.

If the angle between the velocity vector and the magnetic field vector is 0 degree, then force is zero.

When the electrons enters in the magnetic field at any arbitrary angle, it experiences a force and hence it accelerate up.

The change in momentum of the ball is 24 kg·m/s.

The change in momentum of the ball can be found by subtracting the initial momentum from the final momentum. The initial momentum is given by the formula p1 = m * v1, where m is the mass of the ball and v1 is its initial velocity. In this case, the initial velocity is 12 m/s downward, so the initial momentum is -12 kg·m/s. The final momentum is given by the formula p2 = m * v2, where v2 is the velocity of the ball after the bounce. Since the motion after the bounce is the mirror image of the motion before the bounce, the final velocity is 12 m/s upward. Thus, the final momentum is 12 kg·m/s. Subtracting the initial momentum from the final momentum, we get the change in momentum to be 24 kg·m/s.

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The change in momentum of a 1.0-kg ball that bounces with a velocity of 12 m/s upward after striking the ground with a velocity of 12 m/s downward is 24 kg·m/s. This is calculated using the formula for momentum, p = mv, and finding the difference between initial and final momentum.

Momentum (old{p}old{) is calculated by the formula:

p = mv

Step-by-Step Explanation:

The initial momentum just before striking the ground (p₁) is:

p₁ = mass × velocity

p₁ = 1.0 kg × (-12 m/s)

p₁ = -12 kg·m/s

The final momentum just after bouncing up (p₂) is:

p₂ = mass × velocity

p₂ = 1.0 kg × 12 m/s

p₂ = 12 kg·m/s

The change in momentum (Δp) is:

Δp = p₂ - p₁

Δp = 12 kg·m/s - (-12 kg·m/s)

Δp = 12 kg·m/s + 12 kg·m/s

Δp = 24 kg·m/s

So, the change in momentum of the ball is 24 kg·m/s.

Answer:

The body will strike at angle 60.46°

Explanation:

Vertical motion of body:

 Initial speed, u = 40sin30 = 20m/s

 Acceleration, a = 9.81 m/s²

 Displacement, s = 170 m

 We have equation of motion, v² = u² + 2as

 Substituting

   v² = 20² + 2 x 9.81 x 170

    v = 61.12 m/s

 Final vertical speed = 61.12 m/s

 Final horizontal speed = initial horizontal speed = 40cos30= 34.64m/s

 Final velocity = 34.64 i - 61.12 j m/s

 Magnitude

     [tex]v=\sqrt{34.64^2+(-61.12)^2}=70.25m/s[/tex]

 Direction

      [tex]\theta =tan^{-1}\left ( \frac{-61.12}{34.64}\right )=-60.46^0[/tex]

 The body will strike at angle 60.46°

The rotational kinetic energy of the pitcher's arm when throwing a softball at a speed of 139 km/h is calculated using the equation for rotational kinetic energy, and taking moment of inertia and angular velocity into account. The angular velocity is inferred from the linear speed of the ball and the distance it leaves the pitcher's hand from the pivot at the shoulder. The rotational kinetic energy is found to be approximately 1491 Joules.

This question pertains to the rotational kinetic energy of the pitcher's arm during the act of throwing a softball. By definition, the kinetic energy associated with rotational motion (rotational kinetic energy) can be given by the equation: K_rot = 0.5 * I * ω² where 'I' is the moment of inertia and 'ω' is the angular velocity.

To calculate the angular velocity 'ω', we can infer it from the linear speed of the ball when it leaves the pitcher's hand, as 'ω = v/r', v = 139 km/h = 38.6 m/s, and r = 0.600 m (distance ball leaves hand from pivot at shoulder). Therefore, 'ω' is approximately 64.4 rad/s.

Substituting 'I' and 'ω' into the equation above, we get K_rot = 0.5 * 0.720 kg*m² * (64.4 rad/s)² = 1491 Joules, which is the rotational kinetic energy of the pitcher's arm.

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Answer:

(a) 0 rad

(b) 4 rad/s

(c) 28 rad/s

(d)  12 rad/s^2

(e) 6 rad/s^2

(f) 18 rad/s^2  

Explanation:

[tex]\theta (t)=4t-3t^{2}+t^{3}[/tex]   .... (1)

(a) here, we need to find angular displacement when t = 0 s

Put t = 0 in equation (1), we get

[tex]\theta (t=0)=0[/tex]

(b) Angular velocity is defined as the rate of change of angular displacement.

ω = dθ / dt

So, differentiate equation (1) with respect to t.

[tex]\omega =\frac{d\theta }{dt}=4-6t+3t^{2}[/tex]   .... (2)

Angular velocity at t = 2 s

Put t = 2 s in equation (2), we get

ω = 4 - 6 x 2 + 3 x 4 = 4 rad/s

(c) Angular velocity at t = 4 s

Put t = 4 s in equation (2), we get

ω = 4 - 6 x 4 + 3 x 16 = 4 - 24 + 48 = 28 rad/s

(d) Average angular acceleration,

[tex]\alpha =\frac{\omega (t=4s)-\omega (t=2s)}{4-2}[/tex]

α = (28 - 4) / 2 = 12 rad/s^2

(e) The rate of change of angular velocity is called angular acceleration.

α = dω / dt

α = - 6 + 6 t

At t = 2 s

α = - 6 + 12 = 6 rad/s^2

(f) At t = 4 s

α = - 6 + 24 = 18 rad/s^2

Answer:

The ball must be tossed to a height of 4 times the initial height H

Explanation:

We have equation of motion S = ut + 0.5at²

A juggler tosses balls vertically to height H.

That is

      H = 0 x t + 0.5 x a x t²

      H = 0.5at²

To what height must they be tossed if they are to spend twice as much time in the air.

      H' = 0 x 2t + 0.5 x a x (2t)²

      H' = 2at² = 4 H

So the ball must be tossed to a height of 4 times the initial height H

Answer:

[tex]v = 1.11 \times 10^3 m/s[/tex]

Explanation:

By energy conservation law we will have

[tex]U_i + KE_i = U_f + KE_f[/tex]

as we know that as the projectile is rising up then due to gravitational attraction of earth it will slow down

At the highest position the speed of the projectile will become zero

So here we will have

[tex]-\frac{GMm}{R} + \frac{1}{2}mv^2 = -\frac{GMm}{R+h} + 0[/tex]

[tex]-\frac{(6.67 \times 10^{-11})(5.97 \times 10^{24})(100)}{6.37 \times 10^6} + \frac{1}{2}(100)v^2 = -\frac{(6.67 \times 10^{-11})(5.97 \times 10^{-11})(100)}{(6.37 \times 10^{6} + 65000)}[/tex]

[tex] - 6.25 \times 10^7 + 0.5 v^2 = -6.19 \times 10^7[/tex]

[tex]v = 1.11 \times 10^3 m/s[/tex]

Answer:

The inductance is 0.021 H.

Explanation:

Given that,

Number of turns = 16

Current = 3 A

Total flux [tex]\phi=4\times10^{-3}\ Tm^2[/tex]

We need to calculate the inductance

Using formula of total flux

[tex]N\phi = Li[/tex]

[tex]L=\dfrac{N\phi}{i}[/tex]

Where, i = current

N = number of turns

[tex]\phi[/tex] = flux

Put the value into the formula

[tex]L=\dfrac{16\times4\times10^{-3}}{3}[/tex]

[tex]L=0.021\ H[/tex]

Hence, The inductance is 0.021 H.

Answer:

[tex]log_28=3[/tex]

Explanation:

To calculate the value of [tex]log_28[/tex]

Also,Some relation for log values:

[tex]loga^m= mloga[/tex]  -----------------------------------------------------1

[tex]log_aa= 1[/tex]---------------------------------------------------------------2

Also, [tex]2^3= 8[/tex]

Applying in the question as:

[tex]log_28=log_2(2^3)[/tex]

Using Relation-1 mentioned above:

[tex]log_28=3\times log_2(2)[/tex]

Using Relation-2 mentioned above:

[tex]log_28=3\times 1[/tex]

Thus,

[tex]log_28=3[/tex]

To find the electric field in the wire, we first find the cross-sectional area of the wire and then substitute the given values and the calculated area into the formula for electric field. The electric field in the gold wire is approximately 8.24 kN/C.

The question asks for the electric field in a gold wire of given dimensions and current. We'll solve this using the formula for electric field (E) in terms of resistivity (ρ),current (I) and area (A), which is given by E = ρI/A.

First, we need to find the cross-sectional area (A) of the wire using the formula for the area of a circle, since the wire is cylindrical. The area of a circle is given by A = π*(d/2)^2, where d is the diameter. Substituting d = 1.8 mm or 1.8 * 10^-3 m, we find A ≈ 2.54 * 10^-6 m^2.

Next, we need to substitute the known values into the formula E = ρI/A. Using the given values ρ = 2.44 *10^-8 Ω•m, I = 860 mA or 860 * 10^-3 A, and the calculated A ≈ 2.54 * 10^-6 m^2, we find E ≈ 8.24 * 10^3 N/C or 8.24 kN/C.

So, the electric field in the wire is approximately 8.24 kN/C.

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The electric field within the gold wire can be calculated utilizing resistivity formula, determining the cross-sectional area of the wire, and then computing the current density. After these intermediate calculations, the ultimate electric field in the wire is found to be approximately 72 mV/m.

The electric field in the gold wire can be calculated by using the formula for

resistivity: ρ=E/J where

ρ (rho) is the resistivity of the material,

E is the electric field, and

J is the current density.

Before we apply this formula, let's calculate the cross-sectional area of the wire (A) using the formula A=πr², where r is the radius of the wire. The diameter is given as 1.8mm, therefore

r=0.9mm = 0.9x10^-3 m.

Substituting this into the formula gives us

A=π(0.9x10^-3)²≈2.54x10^-6 m².

Now we calculate current density: J=I/A. where I is current, given as

860mA=860x10^-3 A.

Substituting values of I and A into the formula gives

J=860x10^-3/2.54x10^-6≈3.39x10^8 A/m².

Finally, the electric field E=ρ/J=2.44x10^-8/3.39x10^8 ≈ 7.2x10^-2 V/m or 72mV/m.

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The center of mass is given by:

∑mx/∑m

m is the mass of each object

x is the position of each object

We will assign x = 0m to the 18kg mass, therefore x = 6m for the 6kg mass.

∑mx/∑m = (18×0+6×6)/(18+6) = 1.5m

The center of mass is located 1.5m away from the 18kg mass.

The total moment of inertia of the system about the center of mass is given by:

I = ∑mr²

I is the moment of inertia

m is the mass of each object

r is the distance of each object from the center of mass

We know r = 1.5m for the 18kg mass and the rod is 6m long, therefore the 6kg mass must be r = 4.5m from the center of mass.

I = 18(1.5)² + 6(4.5)²

I = 162kg×m²