1.A windmill blade turns counterclockwise and is slowing at a steady rate. Counterclockwise is taken as positive. The angular velocity and the angular acceleration of the blade are

1. a negative angular velocity and a negative angular acceleration
2. a positive angular velocity and a negative angular acceleration
3. a positive angular velocity and a positive angular acceleration
4. a negative angular velocity and a positive angular acceleration

2.A drill bit spins at a steady $40$ rad/s. The time it needs to sweep through $8.0$ rad is

1. $5.0$ s
2. $320$ s
3. $1.26$ s
4. $0.20$ s

3.A centrifuge rotor slows at a steady rate from $90$ rad/s to $30$ rad/s while it turns through $120$ rad. Take the sense of the spin as positive. Its angular acceleration is

1. $-30$ rad/s$^2$
2. $-60$ rad/s$^2$
3. $-0.50$ rad/s$^2$
4. $+30$ rad/s$^2$

4.For a turning disk, the graph of angular velocity against time is a straight line. It starts at a positive value and reaches zero at $t = 4.0$ s. During those first $4.0$ s the disk

1. turns at a steady angular velocity right up until it stops at $t = 4.0$ s
2. has an angular acceleration that falls away to zero as the graph reaches the axis
3. turns in one sense the whole time, with the angle it has swept still growing
4. reverses the sense of its turning at some instant before $t = 4.0$ s

5.Two wheels start from rest, each with a steady angular acceleration. The first has twice the angular acceleration of the second. Each is checked once it has turned through the same angle. At that moment,

1. the two wheels have the same angular velocity and they reached that angle at the same time
2. the first wheel has the greater angular velocity, and it reached that angle in less time
3. the two wheels have the same angular velocity, because they turned through the same angle
4. the second wheel has the greater angular velocity, because it took longer to get there

6.A circular saw blade of radius $0.090$ m spins at $300$ rad/s. The speed of one tooth at the edge of the blade is

1. $54$ m/s
2. $3300$ m/s
3. $300$ m/s
4. $27$ m/s

7.A winch drum of radius $0.25$ m winds in cable at a steady $8.0$ rad/s and lifts a crate. The time the crate needs to rise $6.0$ m is

1. $0.75$ s
2. $0.19$ s
3. $3.0$ s
4. $1.5$ s

8.A bead is clamped to a spoke of a wheel that turns about its axle. For that bead, the rule that its linear speed equals its distance from the axle times the angular velocity holds only while the bead

1. stays put at one place along the spoke as the wheel turns
2. sits out at the rim rather than part way along the spoke
3. turns with a steady angular velocity and no angular acceleration
4. sweeps through an angle that is measured in degrees, not radians

9.On a turning platform a point $0.60$ m from the axis moves at $3.0$ m/s. A second point on the same platform moves at $1.0$ m/s. That second point sits a distance from the axis of

1. $0.60$ m
2. $0.20$ m
3. $1.8$ m
4. $0.40$ m

10.A disk slows at a steady rate. Follow one point on its rim as the disk slows. The two parts of the acceleration of that point behave so that

1. the centripetal part holds a fixed magnitude while the tangential part falls off
2. both parts fall off together and hold a fixed ratio to each other throughout
3. the tangential part turns around to point toward the axis once the disk slows
4. the tangential part holds a fixed magnitude while the centripetal part falls off

11.A child pushes perpendicular to a revolving door panel with a force of $40$ N, at a point $0.80$ m from the central shaft. The torque about the shaft is

1. $32$ N$\cdot$m
2. $40$ N$\cdot$m
3. $50$ N$\cdot$m
4. $0.020$ N$\cdot$m

12.A horizontal bar turns freely about a pivot at its left end. A force pushes straight down on the bar at its right end. Counterclockwise is taken as positive. The torque of that force about the pivot is

1. zero, because the force is perpendicular to the bar
2. zero, because the pivot holds that end of the bar in place
3. negative, because acting alone it would turn the bar clockwise
4. positive, because the force acts right at the far end of the bar

13.A cyclist pushes straight down on a bicycle pedal with a force of $150$ N. The crank arm is $0.18$ m long and makes an angle of $60$ degrees with the vertical. The torque about the crank axle is

1. $46.8$ N$\cdot$m
2. $23.4$ N$\cdot$m
3. $27.0$ N$\cdot$m
4. $13.5$ N$\cdot$m

14.A rope pulls on a gate at a point $2.0$ m from the hinge, in a direction the gardener may choose. Together with the magnitude of the pull, the one quantity that fixes the torque about the hinge is

1. the distance from the hinge to the point of contact, whatever way the rope points
2. the full length of the gate, measured from the hinge out to its far edge
3. the angle the rope makes with the gate, with no distance needed at all
4. the perpendicular distance from the hinge to the line of the rope

15.One force acts on a bar at a single point. The axis that torques are taken about is then moved to a new place. Moving the axis

1. changes the torque only while the new axis lies somewhere along the bar itself
2. reverses the sign of the torque for every new axis that might be chosen
3. changes the torque, and the torque is zero for any axis lying on the line of the force
4. leaves the torque unchanged, because the force acting on the bar has not changed at all

16.A $0.50$ kg ball sits at the far end of a light arm, $0.80$ m out from the axle it spins around. Its rotational inertia about that axle is

1. $0.32$ kg$\cdot$m$^2$
2. $0.40$ kg$\cdot$m$^2$
3. $0.64$ kg$\cdot$m$^2$
4. $0.16$ kg$\cdot$m$^2$

17.Two small masses are fixed to a light frame that turns about a fixed axis: $3.0$ kg at $0.20$ m from the axis and $1.0$ kg at $0.60$ m from the axis. The rotational inertia of the pair is

1. $1.4$ kg$\cdot$m$^2$
2. $0.48$ kg$\cdot$m$^2$
3. $1.2$ kg$\cdot$m$^2$
4. $0.16$ kg$\cdot$m$^2$

18.Two children of equal mass sit on opposite sides of a light turning platform, each the same distance from the axis. One child then moves in closer to the axis. The rotational inertia of the whole system

1. falls in direct proportion to the distance that one child has moved in
2. rises, because the mass is no longer placed evenly around the axis
3. falls, because the distance of that child from the axis is squared in the sum
4. does not change, because no mass was added to the system and none was taken off it

19.A thin hoop of mass $1.6$ kg and radius $0.30$ m has $I_{\mathrm{cm}} = MR^{2}$ about its central axis. The hoop now hangs on a nail through one point of the hoop and swings in its own plane. About the nail, its rotational inertia is

1. $0.14$ kg$\cdot$m$^2$
2. $0.22$ kg$\cdot$m$^2$
3. $0.58$ kg$\cdot$m$^2$
4. $0.29$ kg$\cdot$m$^2$

20.A rigid system is turned about two parallel axes, one through its center of mass and one a short distance away. Set against the value about the center of mass, the rotational inertia about the other axis is

1. greater, by the mass of the system times the square of the distance between the axes
2. greater, by the mass of the whole system times the distance between the two parallel axes
3. the same, because none of the mass of the system has been moved anywhere at all
4. less, because the second axis sits closer to part of the mass of the system

21.A wheel on good bearings turns at a steady angular velocity, with nothing driving it. Which statement about the wheel is true?

1. The torques on it add to a steady value that is not zero while it turns.
2. It is in rotational equilibrium, because the torques on it add to zero.
3. It is not in rotational equilibrium, because it is turning rather than at rest.
4. It is in rotational equilibrium only at the instant the wheel comes to rest.

22.A light rod hangs level from a string tied at its center. A $0.60$ kg block hangs $0.20$ m to the left of the string. A $0.40$ kg block is hung on the right side. The rod hangs level when that block is

1. $0.20$ m from the string
2. $0.12$ m from the string
3. $0.30$ m from the string
4. $0.13$ m from the string

23.A flat plate rests on frictionless ice. One steady push is applied along a line that passes through the center of mass of the plate. While that push acts, the plate is

1. in translational equilibrium, because the push acts through the center of mass
2. in both kinds of equilibrium, because only one push is applied to the plate
3. in neither kind of equilibrium, because any push at all must start the plate spinning
4. in rotational equilibrium, because the net torque about the center of mass is zero

24.Two brackets hold a uniform shelf of mass $4.0$ kg that is $1.20$ m long, one bracket at the left end and one $0.90$ m to the right of it. A $6.0$ kg box stands $0.50$ m in from the left end. The left bracket pushes up with

1. $27$ N
2. $40$ N
3. $50$ N
4. $60$ N

25.Three forces act on a bar. The net force on the bar is zero, and the net torque about one chosen axis is zero as well. The net torque about any other axis is

1. zero as well, and that follows from the net force on the bar being zero
2. zero only for another axis that also lies somewhere along the bar itself
3. able to take any value at all, since a torque is always tied to its axis
4. zero only for an axis that runs through the center of mass of the bar

26.A revolving stage has a rotational inertia of $18$ kg$\cdot$m$^2$ about its central axis. A net torque of $27$ N$\cdot$m acts on it. The angular acceleration of the stage is

1. $0.67$ rad/s$^2$
2. $486$ rad/s$^2$
3. $2.7$ rad/s$^2$
4. $1.5$ rad/s$^2$

27.A turnstile of rotational inertia $2.4$ kg$\cdot$m$^2$ is pushed with $30$ N perpendicular to an arm, $0.80$ m from the axis. Friction opposes the turning with a torque of $6.0$ N$\cdot$m. The angular acceleration is

1. $12.5$ rad/s$^2$
2. $18$ rad/s$^2$
3. $7.5$ rad/s$^2$
4. $10$ rad/s$^2$

28.A wheel is turning counterclockwise. A net clockwise torque is then applied to it. Over the next short interval the wheel

1. keeps turning counterclockwise and its angular velocity falls off
2. turns clockwise at once, following the sense of the new net torque
3. keeps the angular velocity it had, since the torque opposes the spin
4. stops at that very instant, because the torque opposes the spin

29.A flywheel of rotational inertia $0.80$ kg$\cdot$m$^2$ starts from rest. A steady net torque of $2.0$ N$\cdot$m acts on it for $6.0$ s. The angle it turns through in that time is

1. $2.5$ rad
2. $45$ rad
3. $90$ rad
4. $15$ rad

30.A wheel turns on an axle, and a small pulley is fixed to the same axle. A rope is pulled with the same force, first around the rim of the wheel and then around the small pulley. The angular acceleration is

1. the same in the two cases, because the force and the turning system are the same in both
2. greater with the rope on the small pulley, because the force then acts closer in to the axle
3. greater with the rope at the rim, because the rotational inertia itself falls away as the force moves out
4. greater with the rope at the rim, because the torque is greater and the rotational inertia is the same

31.*A spin dryer basket speeds up from rest at $2.0$ rad/s$^2$ for $5.0$ s, then slows at a steady $1.0$ rad/s$^2$ until it stops. Its angular velocity on reaching half of its total angle is

1. $7.1$ rad/s
2. $10$ rad/s
3. $8.7$ rad/s
4. $5.0$ rad/s

32.*A flywheel is coming to a stop at a steady rate. One spot sits on its rim and a second half way in toward the axle. At one instant, the angle between the total acceleration of a spot and its own radius is

1. the same for the two spots, because the ratio of the two parts does not depend on the radius
2. greater for the spot on the rim, because that spot travels faster along its own circular path
3. greater for the inner spot, because its centripetal part is the weaker of its own two parts
4. impossible to compare until the mass of the flywheel and the rate of the slowing are given

33.*A student claims that the rotational inertia of a rigid body about an axis through its center of mass is less than its value about any other axis whatsoever. That claim

1. fails, because the theorem takes a term away for an axis that runs inside the body
2. fails, because an axis in a new direction through the center of mass can give less
3. holds, because the parallel axis theorem adds a term that is never a negative one
4. holds, because the center of mass is the one point the mass of the body balances about

34.*A uniform diving board of mass $30$ kg and length $4.0$ m is bolted to the deck at its back end and rests on a roller $1.0$ m from that end. A $50$ kg swimmer stands at the free end. The force from the bolt is

1. $1500$ N, directed downward
2. $800$ N, directed upward
3. $2600$ N, directed upward
4. $1800$ N, directed downward

35.*A uniform $20$ kg gate of width $1.2$ m is at rest on vertical hinges, with $I = \tfrac13ML^{2}$ about that line. A child pushes the outer edge with $25$ N, always perpendicular to it. After a $90$ degree swing the angular velocity is

1. $1.8$ rad/s
2. $4.4$ rad/s
3. $3.1$ rad/s
4. $24$ rad/s

1.For a rigid body turning about a fixed axis, explain why the angle is used as the position variable rather than the distance that one point of the body travels.

2.A ceiling fan is switched off and slows to rest at a steady rate. Explain why the fan turns through a greater angle during the first half of the slowing time than during the second half.

3.Explain how a wheel can have an angular velocity of zero at one instant and an angular acceleration that is not zero at that same instant. Describe a case in which this happens.

4.One wheel is reported at $+4.0$ rad/s and another at $-4.0$ rad/s. State what the two wheels have in common and what differs between them, and explain what fixes the sign.

5.On a carousel, one horse is mounted near the axis and another out near the rail. Compare the angular velocities of the two horses and compare their linear speeds, and explain both comparisons.

6.A garden hose is wound onto a reel. Explain why the length of hose taken up in one full turn is fixed by the radius of the coil and not by how fast the reel is turned.

7.Explain why the angle has to be measured in radians before the relation between arc length and angle, or the relation between linear speed and angular velocity, may be used.

8.Describe why the steering wheel of a truck without power steering is built with a large radius, and explain the comparison with a wheel of small radius in terms of torque about the steering column.

9.A meter stick is balanced on a pivot at its middle. One hand presses straight down on the left end and the other hand presses straight down on the right end. Explain why the two torques about the pivot carry opposite signs.

10.A child pulls on the outer rail of a playground merry-go-round, with the rope at $50$ degrees to the line from the axis out to her hands. Explain which part of that pull turns the merry-go-round and what the other part does.

11.A batter slides her hands farther up the handle of a bat, toward the heavy barrel, and swings about her hands. Explain what that does to the rotational inertia of the bat about the axis through her hands.

12.Weights are clamped at several places along a light arm that spins about a fixed axle. Describe how to build up the rotational inertia of the whole group, and explain why a weight clamped close to the axle adds little to it.

13.A door can turn about its hinge line, and the same door can be spun about a line through its center. Explain why one body does not have a single rotational inertia.

14.A tall cabinet stands on a level floor and is pushed gently near its top so that it leans. Explain, in terms of torques about the front bottom edge, why the cabinet tips over once the line of its weight passes beyond that edge.

15.Explain why a tower crane carries a heavy counterweight on the arm opposite its load, and state what has to be true of the torques about the mast while a load hangs from the crane.

16.Describe how to pick the axis for a torque equation in an equilibrium problem so that one unknown force drops out, and explain why picking it that way is allowed.

17.A cart is pulled along a level floor by a cord that winds onto a winch drum turning on a fixed axle. Describe the two bodies that have to be treated separately here, and state the one equation that ties them together.

18.Two bicycle wheels of equal mass are spun up to the same angular velocity and left to slow under the same friction torque at the axle. One wheel carries most of its mass out at the rim. Explain which wheel takes longer to stop.

19.Several torques act on a flywheel at once. Describe how to find its angular acceleration, and explain why the torques have to be added with their signs before anything is divided.

20.Two flywheels start from rest on their axles and are driven by the same net torque for the same length of time. One has twice the rotational inertia of the other. Compare the angles they turn through, and explain the comparison.

21.*A car is fitted with tires of greater radius and nothing else is changed. The speedometer is driven by the angular velocity of a wheel. Explain whether the speedometer now reads above or below the true road speed.

22.*A student claims that a force applied farther out from the axis always produces a torque of greater magnitude than the same force applied closer in. Explain the flaw in that claim.

23.*A uniform rod has a known rotational inertia about an axis at one end. Somebody finds the value about a parallel axis a short distance beyond that end by adding the mass times the square of that short distance. Explain the flaw.

24.*A uniform bar is pinned to a wall at one end and held level by a cable running from its far end up to a higher point on the wall. A student claims the force the pin exerts has to point along the bar. Explain the flaw.

25.*A cord on the rim of a pulley is pulled by a hand with a steady $8$ N. In a second trial a block of weight $8$ N hangs from the same cord instead. A student claims the angular acceleration is the same in both trials. Explain the flaw.