Exercise

Exercises 7.7 Exercise

MI.

1.

A ball, a cylinder, and a ring each of mass 10 kg and radius 0.1 m are released from rest at the top of an inclined plane 2 m long. Which will reach the bottom of the plane the fastest? Why?

2.

A meter stick of mass 500 g is pivoted about a point located at the 75 cm mark. What is the moment of inertia of the meter stick?

3.

A wagon wheel consists of a rim of mass 50 kg and diameter 60 cm and eight spokes each with a mass of 2 kg. What is the moment of inertia of the wheel?

4.

A rectangular block has a length of 20 cm, a width of 10 cm, a height of 5 cm, and a mass of 4 kg. What is its moment of inertia about its three principle axes?

5.

Find the moment of inertia of the following uniform bodies about the axis indicated.

6.

Find the moment of inertia of

a physical pendulum and
a square about the axis indicated.

7.

Find the moment of inertia of an equilateral triangle about one of its vertices and perpendicular to its plane. The triangle is made out of three rods.

Rot Dyn.

8.

A body rotates according to the equation $\theta = At + Bt^{3}\text{,}$ where $A = 10 \,rad/s$ and $B = 2 \,rad/s^{3}\text{.}$ Find:

$\displaystyle \theta(2s)$
$\displaystyle \omega_{av}(0,2s)$
$\displaystyle \omega(3s)$
$\displaystyle \alpha_{av}(0,3s)$
$\displaystyle \alpha(4s)$

9.

An electric motor accelerates uniformly from rest to $600 \,rpm$ in $10 \,s.$

What is its angular acceleration?
How many revolutions does it make?
What was its angular velocity at the end of 2 s?
How fast was it rotating after making 10 revolutions?
How long did it take to make 10 revolutions?

10.

A uniform cylinder of mass 100 kg has a radius of 30 cm and spins freely about its axis of symmetry. One end of a string is attached to the surface of the cylinder so that the string winds and unwinds as the cylinder rotates first one direction and then the other. A constant tension of 100 N is maintained in the string by a constant vector force applied to the other end of the string. At time t = 0, the string is completely unwound and the cylinder is spinning at 30 rad/s.

What is the moment of inertia of the cylinder?
As long as $\theta \gt \frac{\pi}{2},$ or $\theta \lt -\frac{\pi}{2}\text{,}$ the torque on the cylinder is constant in magnitude. Why? What is this torque?
Ignoring the first quarter revolution, how long does it take the cylinder to stop?
What is the angular acceleration of the cylinder when it is slowing down?
What is the minimum length of string that will stop the cylinder?
What is the initial kinetic energy of the cylinder?
What is the initial angular momentum of the cylinder?
If the cylinder is rotating counter-clockwise, what is the direction of the angular momentum vector?

11.

An Atwood machine as shown at right consists of a mass of 10 kg supported by a rope draped over a uniform cylinder of mass 20 kg and radius 10 cm and then tied to a mass of 5 kg located 1.5 m below the first mass. If the system is released from rest,

find the tension $T_{1}$ in the rope supporting the $10-kg$ mass.
What is the maximum height ever attained by the second mass.

12.

A thin ring of mass 5 kg and diameter 1 m is pivoted about a point on its rim. If the system is released from rest with the ring directly above the pivot point as shown in figure and if it swings downward, when it is exactly half way down to the bottom find

how fast it is moving,
the radial and tangential acceleration of its center of mass, and
the horizontal and vertical forces exerted on the ring by its pivot.

13.

A 186.8 N sphere with radius R= 101.6 mm is placed on a horizontal surface with initial angular velocity $\omega_{o} = 40 \,rad/s.$ The coefficient of kinetic friction between the sphere and the surface is $\mu_{k} = 0.06.$ What maximum velocity will the center of the sphere attain, and how long does it take to reach that velocity?

14.

A 1 m long and 2 kg stick is nailed to the wall so that it can rotate freely about the end. A 1 kg ball, with speed 3 m/s hits with the stick at some distance $x$ below the pivot point. The ball collides elastically with the stick, and stops dead after collision. Find the stick’s resulting initial angular velocity. Find the distance $x.$

15.

A 25 g bug crawls from the center to the outside edge of a 150 g disk of radius 15.0 cm. The disk was rotating at 15.0 rad/s. What will be its final angular velocity? Treat the bug as a point mass.

16.

A bicycle tire has a mass of 4.0 kg and a radius of 0.33 m. If it is rotating at 22 rad/s what is its angular momentum? If it is used as a gyroscope with a 24 cm long pivot bar, what will be its precession speed?

17.

Calculate the angular momentum of a phonograph record, LP (Long Play) rotating at $33\frac{1}{3}$ rev/min. An LP has a radius of 15 cm and a mass of 150 g. A typical phonograph can accelerate an LP from rest to its final speed in 0.35 s, what average torque would be exerted on the LP?

Osc_Motion.

18.

What are the simple harmonic motion equations for a particle starting out from rest at $x_{o}$ when $t = 0\text{?}$

19.

What are the simple harmonic motion equations for a particle passing through $x = 0$ with velocity $v_{o}$ when $t = 0\text{?}$

20.

The piston in the engine of an automobile oscillates at 1000 rpm in simple harmonic motion of amplitude 5 cm. If its mass is 0.25 kg, find

its period of oscillation,
its angular frequency,
its maximum displacement,
its maximum velocity,
its maximum acceleration,
the maximum force exerted on the piston.

21.

A simple harmonic oscillator of mass 10 kg on a spring of constant 100 N/m starts out at x = 10 cm moving in the +x direction with a velocity of 10 m/s. Find

$\displaystyle \omega$
$\displaystyle A,$
$\displaystyle \theta$
$\displaystyle T,$
$\displaystyle x(1s),$
$\displaystyle v(1s),$
$\displaystyle a(1s),$
$\displaystyle K(1s),$
$\displaystyle U(1s),$
$\displaystyle E(1s),$
$\displaystyle v_{max}$
$\displaystyle a_{max}$

22.

A pendulum of mass 2 kg and length 2 m is displaced 10 cm from its equilibrium position and released.

What is the equation giving the horizontal force F(x) required to displace the mass a horizontal distance x for small values of x?
Is this equation valid for x = 10 cm? Why or why not?
What is the effective spring constant of the pendulum?
What is the period of the pendulum?
What is the equation of motion for this pendulum?
What is the kinetic energy of the pendulum when it is 5 cm from the equilibrium position?

23.

A mass of 1 kg is hung vertically from a spring stretching the spring 20 cm. It is then pulled 10 cm below its equilibrium position and released from rest. Find

the angular frequency of oscillation
the spring constant of the spring,
the amplitude of oscillation,
the kinetic energy of mass when passing through equilibrium,
the potential energy of the spring when the mass passes through equilibrium,
the time to first reach equilibrium,
the time to make one complete cycle.

24.

A 250 gram bob is hanging on a 80 cm long string. Its velocity is 0.25 m/s when observed at mean position. What maximum angle does it reach?

Hint.

\begin{equation*} \theta_{max}=\frac{x_{max}}{L}, \end{equation*}

\begin{equation*} x_{max} =A = \frac{v_{max}}{\omega} \end{equation*}

25.

Two pendula have strings of equal length but the mass of the second pendulum is four times the mass of the first pendulum. If $f_{1}$ is the frequency of the first pendulum and $f_{2}$ is the frequency of the second pendulum, then determine the relationship between $f_{1}$ and $f_{2}\text{.}$

26.

Determine the effect on time period of a simple pendulum if the iron bob is replaced by a wooden one of same mass? Neglect air resistance.

27.

A simple pendulum has a length L and a time period T on the surface of the earth. What would be its time period when it is taken to the surface of a planet of acceleration due to gravity is $g/4?$

28.

A block of mass 2 kg is attached to the spring - mass system of spring constant 200 N/m. A system is executing SHM and has velocity 40 m/s at 3 m away from equlibrium position. Determine its amplitude of oscillation.

Prev Top Next