Chapter 9

A roller in a printing press turns through an angle \(\theta(t)\) given by \(\theta(t)=\gamma t^{2}-\beta t^{3},\) where \(\gamma=3.20 \operatorname{rad} / \mathrm{s}^{2}\) and \(\beta=\) 0.500 \(\mathrm{rad} / \mathrm{s}^{3}\) , (a) Calculate the angular velocity of the roller as a function of time. (b) Calculate the angular acceleration of the roller as a function of time. (c) What is the maximum positive angular velocity, and at what value of \(t\) does it occur?

When a toy car is rapidly scooted across the floor, it stores energy in a flywheel. The car has mass 0.180 \(\mathrm{kg}\) , and its flywheel has moment of inertia \(4.00 \times 10^{-5} \mathrm{kg} \cdot \mathrm{m}^{2} .\) The car is 15.0 \(\mathrm{cm}\) long. An advertisement claims that the car can travel at a scale speed of up to 700 \(\mathrm{km} / \mathrm{h}(440 \mathrm{mi} \mathrm{h}) .\) The scale speed is the speed of the toy car multiplied by the ratio of the length of an actual car to the length of the toy. Assume a length of 3.0 \(\mathrm{m}\) for a real car. (a) For a scale speed of 700 \(\mathrm{km} / \mathrm{h}\) , what is the actual translational speed of the car? (b) If all the kinetic energy that is initially in the flywheel is converted to the translational kinetic energy of the toy, how much energy is originally stored in the flywheel? (c) What initial angular velocity of the flywheel was needed to store the amount of energy calculated in part (b)?

A classic 1957 Chevrolet Corvette of mass 1240 \(\mathrm{kg}\) starts from rest and speeds up with a constant tangential acceleration of 3.00 \(\mathrm{m} / \mathrm{s}^{2}\) on a circular test track of radius 60.0 \(\mathrm{m}\) . Treat the car as a particle. (a) What is its angular acceleration? (b) What is its angular speed 6.00 s after it starts? (c) What is its radial acceleration at this time? (d) Sketch a view from above showing the circular track, the car, the velocity vector, and the acceleration component vectors 6.00 s after the car starts. (e) What are the magnitudes of the total acceleration and net force for the car at this time? (f) What angle do the total acceleration and net force make with the car's velocity at this time?

Engineers are designing a system by which a falling mass \(m\) imparts kinetic energy to a rotating uniform drum to which it is attached by thin, very light wire wrapped around the rim of the drum (Fig. 9.34\()\) . There is no appreciable friction in the axle of the drum, and everything starts from rest. This system is being tested on earth, but it is to be used on Mars, where the acceleration due to gravity is 3.71 \(\mathrm{m} / \mathrm{s}^{2} .\) In the earth tests, when \(m\) is set to 15.0 \(\mathrm{kg}\) and allowed to fall through \(5.00 \mathrm{m},\) it gives 250.0 \(\mathrm{J}\) of kinetic energy to the drum. (a) If the system is operated on Mars, through what distance would the \(15.0-0\) mass have to fall to give the same amount of kinetic energy to the drum? (b) How fast would the 15.0 \(\mathrm{kg}\) mass be moving on Mars just as the drum gained 250.0 \(\mathrm{J}\) of kinetic energy?

In. A vacuum cleaner belt is looped over a shaft of radius 0.45 \(\mathrm{cm}\) and a wheel of radius 2.00 \(\mathrm{cm} .\) The arrangement of the bell, shaft, and wheel is similar to that of the chain and sprockets in Fig. 9.14 . The motor turns the shaft at 60.0 rev/s and the moving belt turns the wheel, which in turn is connected by another shaft to the roller that beats the dirt out of the rug being vacuumed. Assume that the belt doesn't ship on either the shaft or the wheel. (a) What is the speed of a point on the belt? (b) What is the angular velocity of the wheel, in rad/s?

The motor of a table saw is rotating at 3450 rev/min. A pulley attached to the motor shaft drives a second pulley of half the diameter by means of a V-belt. A circular saw blade of diameter 0.208 \(\mathrm{m}\) is mounted on the same rotating shaft as the second pulley. (a) The operator is careless and the blade catches and throws back a small piece of wood. This piece of wood moves with linear speed equal to the tangential speed of the rim of the blade. What is this speed? (b) Calculate the radial acceleration of points on the outer edge of the blade to see why sawdust doesn't stick to its teeth.

A wheel changes its angular velocity with a constant angular acceleration while rotating about a fixed axis through its center (a) Show that the change in the magnitude of the radial acceleration during any time interval of a point on the wheel is twice the product of the angular acceleration, the angular displacement, and the perpendicular distance of the point from the axis. (b) The radial acceleration of a point on the wheel that is 0.250 \(\mathrm{m}\) from the axis changes from 25.0 \(\mathrm{m} / \mathrm{s}^{2}\) to 85.0 \(\mathrm{m} / \mathrm{s}^{2}\) as the wheel rotates through 15.0 rad. Calculate the tangential acceleration of this point. (c) Show that the change in the wheel's kinetic energy during any time interval is the product of the moment of inertia about the axis, the angular acceleration, and the angular displacement. (d) During, the 15.0 -rad angular displacement of part (b), the kinetic energy \(y\) y of the wheel increases from 20.0 \(\mathrm{J}\) to 45.0 \(\mathrm{J}\) . What is the moment of inertia of the wheel about the rotation axis?

A sphere consists of a solid wooden ball of uniform density 800 \(\mathrm{kg} / \mathrm{m}^{3}\) and radius 0.20 \(\mathrm{m}\) and is covered with a thin coating of lead foil with area density 20 \(\mathrm{kg} / \mathrm{m}^{2} .\) Calculate the moment of inertia of this sphere about an axis passing through its center.

Estimate your own moment of inertia about a vertical axis through the center of the top of your head when you are standing up straight with your arms outstretched. Make reasonable approximations and measure or estimate necessary quantities.

A thin uniform rod 50.0 \(\mathrm{cm}\) long with mass 0.320 \(\mathrm{kg}\) is bent at its center into a \(\mathrm{V}\) shape, with a \(70.0^{\circ}\) angle at its vertex. Find the moment of inertia of this \(\mathrm{V}\) -shaped object about an axis perpendicular to the plane of the \(\mathrm{V}\) at its vertex.

It has been argued that power plants should make use of off-peak hours (such as late at night) to generate mechanical energy and store it until it is needed during peak load times, such as the middle of the day. One suggestion has been to store the energy in large flywheels spinning on nearly frictionless ball bearings. Consider a flywheel made of iron (density 7800 \(\mathrm{kg} / \mathrm{m}^{3} )\) in the shape of a \(10.0-\mathrm{cm}\) -hick uniform disk. (a) What would the diameter of such a disk need to be if it is to store 10.0 megajonles of kinetic energy when spinning at 90.0 \(\mathrm{rpm}\) about an axis perpendicular to the disk at its center? (b) What would be the centripetal acceleration of a point on its rim when spinning at this rate?

While redesigning a rocket engine, you want to reduce its weight by replacing a solid spherical part with a hollow spherical shell of the same size. The parts rotate about an axis through their center You need to make sure that the new part always has the same rotational kinetic energy as the original part had at any given rate of rotation. If the original part had mass \(M,\) what must be the mass of the new part?

A uniform, solid disk with mass \(m\) and radius \(R\) is pivoted about a horizontal axis through its center. A small object of the same mass \(m\) is glued to the rim of the disk. If the disk is released from rest with the small object at the end of a horizontal radius, find the angular speed when the small object is directly below the axis.

A metal sign for a car dealership is a thin, uniform right triangle with base length \(b\) and height \(h\) . The sign has mass \(M\) . (a) What is the moment of inertia of the sign for rotation about the side of length \(h ?\) If \(M=5.40 \mathrm{kg}, b=1.60 \mathrm{m},\) and \(h=1.20 \mathrm{m},\) what is the kinetic energy of the sign when it is rotating about an axis along the \(1.20-\mathrm{m}\) side at 2.00 \(\mathrm{rev} / \mathrm{s} ?\)

A meter stick with a mass of 0.160 \(\mathrm{kg}\) is pivoted about one end so it can rotate without friction about a horizontal axis. The meter stick is held in a horizontal position and released. As it swings through the vertical, calculate (a) the change in gravitational potential energy that has occurred; (b) the angular speed of the stick; (c) the linear speed of the end of the stick opposite the axis. (d) Compare the answer in part (c) to the speed of a particle that has fallen \(1.00 \mathrm{m},\) starting from rest.

A passenger bus in Zurich, Switzerland, derived its motive power from the energy stored in a large flywheel. The wheel was brought up to speed periodically, when the bus stopped at a station, by an electric motor, which could then be attached to the electric power lines. The flywheel was a solid cylinder with mass 1000 \(\mathrm{kg}\) and diameter \(1.80 \mathrm{m} ;\) its top angular speed was 3000 \(\mathrm{rev} / \mathrm{min}\) . (a) At this angular speed, what is the kinetic energy of the flywheel? (b) If the average power required to operate the bus is \(1.86 \times 10^{4} \mathrm{W},\) how long could it operate between stops?

A thin, flat, uniform disk has mass \(M\) and radius \(R\) . A circular hole of radius \(R / 4,\) centered at a point \(R / 2\) from the disk's center, is then punched in the disk. (a) Find the moment of inertia of the disk with the hole about an axis through the original center of the disk, perpendicular to the plane of the disk. (Hint: Find the moment of inertia of the piece punched from the disk. (b) Find the moment of inertia of the disk with the hole about an axis through the center of the hole, perpendicular to the plane of the disk.

A pendulum is made of a uniform solid sphere with mass \(M\) and radius \(R\) suspended from the end of a light rod. The distance from the pivot at the upper end of the rod to the center of the sphere is \(L\) . The pendulum's moment of inertia I for rotation about the pivot is usually approximated as \(M L^{2} .\) (a) Use the parallel-axis theorem to show that if \(R\) is 5\(\%\) of \(L\) and the mass of the rod is ignored, \(I_{p}\) is only 0.1\(\%\) greater than \(M L^{2} .\) (b) If the mass of the rod is 1\(\%\) of \(M\) and \(R\) is much less than \(L,\) what is the ratio of \(I_{\text { rod }}\) for an axis at the pivot to \(M L^{2} ?\)

A thin, uniform rod is bent into a square of side length \(a\) . If the total mass is \(M\) , find the moment of inertia about an axis through the center and perpendicular to the plane of the square. (Hint: Use the parallel-axis theorem.)

A cylinder with radius \(R\) and mass \(M\) has density that increases linearly with distance \(r\) from the cylinder axis, \(\rho=\alpha r\) where \(\alpha\) is a positive constant. (a) Calculate the moment of inertia of the cylinder about a longitudinal axis through its center in terms of \(M\) and \(R .\) (b) Is your answer greater or smaller than the moment of inertia of a cylinder of the same mass and radius but of uniform density? Explain why this result makes qualitative sense.

The moment of inertia of a sphere with uniform density about an axis through its center is \(\frac{2}{5} M R^{2}=0.400 M R^{2} .\) Satellite observations show that the earth's moment of inertia is 0.3308\(M R^{2}\) . Geophysical data suggest the earth consists of flve main regions: the inner core \((r=0 \text { to } r=1220 \mathrm{kn})\) of average density 12, \(900 \mathrm{kg} / \mathrm{m}^{3},\) the outer core \((r=1220 \mathrm{kin} \text { to } r=3480 \mathrm{kin})\) of average density \(10,900 \mathrm{kg} / \mathrm{m}^{3},\) the lower mantle \((r=3480 \mathrm{kn} \text { to }\) \(r=5700 \mathrm{kin}\) of average density 4900 \(\mathrm{kg} / \mathrm{m}^{3}\) , the upper mantle \((r=5700 \mathrm{kn} \text { to } r=6350 \mathrm{kn})\) of average density 3600 \(\mathrm{kg} / \mathrm{m}^{3}\) and the outer crust and oceans \((r=6350 \mathrm{km} \text { to } r=6370 \mathrm{kn})\) of average density 2400 \(\mathrm{kg} / \mathrm{m}^{3}\) . (a) Show that the moment of inertia about a diameter of a uniform spherical shell of inner radius \(R_{1}\) . outer radius \(R_{2}\) , and density \(\rho\) is \(I=\rho(8 \pi / 15)\left(R_{2}^{5}-R_{1}^{5}\right) .\) (Hint: Form the shell by superposition of a sphere of density \(\rho\) and a smaller sphere of density \(-\rho .\) (b) Check the given data by using them to calculate the mass of the earth. (c) Use the given data to calculate the earth's moment of inertia in terms of \(M R^{2}\) .

Calculate the moment of inertia of a uniform solid cone about an axis through its center (Fig. 9.40). The cone has mass \(M\) and altitude \(h\) . The radius of its circular base is \(R\) . figure can't copy