Angular Momentum and Rolling Motion
Torque/Static Equilibrium
Rotational Kinetics
Moments of Inertia
Test Your Luck
100
A bicycle tire of radius 0.33 m moves forward, without slipping, a distance of 0.27 m. How many degrees does the wheel rotate?
47 degrees
100
When you open a door, why do you push as far away from the door hinges (axis of rotation) and as perpendicular to the surface as you can?
Torque is dependent upon the distance between the force application and the object. It also depends on the angle that it makes with a line connecting the force to the object. Both are maximized by pushing at the far end of the door in a perpendicular fashion, resulting in a greater angular acceleration
100
A tricycle wheel of radius 0.11 m is at rest and is then accelerated at a rate of 2.3 rad/s2 for a period of 8.6 s. What is the wheel’s final angular velocity?
20 rad/s
100
When does an ice skater have the greatest moment of inertia: when their arms are outreached or when their arms are brought in?
When their arms are outreached
100
What is the physics term for the eye of the storm? Why would winds be weaker at the eye of the tornado than at its outermost edge?
The eye of the storm is the center of rotation. Winds are weaker at the eye of a tornado because tangential velocity is directly proportional to the radius of curvature.
200
A cockroach with mass m rides on a disk of mass 6m and radius R. The disk rotates like a merry-go-round around its central axis at the angular speed wi =1.5 rad/s. The cockroach is initially at radius r = 0.8R, but then it crawls out to the rim of the disk. Treat the cockroach as a particle. As the cockroach crawls out to the rim does angular momentum increase, decrease, or stay the same?
Stays the same
200
If you have a flat tire, and you’re using a wrench to loosen the nuts that hold the tire rim to the axel, is it advantageous to have a longer wrench or a shorter one? Why?
A longer wrench will maximize the torque due to a given force (your strength) and will result in a greater angular acceleration of the wheel nuts. The goal is to provide a greater torque than what currently is holding the nuts in place – friction between the nut and the wheel rim and bolt.
200
A tricycle wheel of radius 0.11 m is at rest and is then accelerated at a rate of 2.3 rad/s2 for a period of 8.6 s. What is the wheel’s final linear velocity?
2.2 m/s
200
You are stepping on a merry-go-round with two rings of fiberglass horses – an inner ring and an outer ring. You get motion sick very easily. Should you choose a horse in the inner or outer ring to ride? Why?
Both rings are moving with the same angular acceleration and velocity. However, the further you are away from the center of the merry-go-round, the greater the tangential and centripetal acceleration and the velocity. This is what you feel, so if you are prone to motion sickness, you want slower magnitudes of the linear quantities. You should choose the inner ring.
200
What is the formula for the kinetic energy of rolling motion?
K=1/2Mv2com+1/2Iω2
300
A ladybug and grasshopper are on a disk. The grasshopper is at a distance of 0.2R and the ladybug is at a distance of R. Which insect experiences the greater linear velocity, or are they the same?
The ladybug experiences the greater linear velocity
300
A scaffold of mass mS=60kg and length L = 5 m is supported in a horizontal position by a vertical cable at each end. A window washer of mass mw=80kg stands at a point 1.5 m from one end. Goal: determine the tension in the near cable Tnear and the tension in the far cable Tfar
Tfar = 529 N; Tnear = 843 N
300
A solid cylinder rotates with constant angular acceleration about a fixed axis. The cylinder’s moment of inertia (I) about the axis is 4.0 kg*m2. At time t = 0 s, the cylinder is at rest. At time, t = 2.0 s, its angular velocity is 4.0 rad/s. What is the angular acceleration?
2.0 rad/s2
300
What torque needs to be applied to a hula hoop of radius 0.58 m and mass 2.5 kg (model it as a hoop, with I = MR2) to give it an angular acceleration of 6.8 rad/s2?
5.7N*m
300
What is the rotational kinetic energy of a 0.82 kg sphere (I = 2/5 MR2), with a radius of 0.058 m, rolling with an angular velocity of 5.2 rad/s?
0.015 J
400
A hollow cylinder of mass M and radius R rolls smoothly down an inclined plane. A block of mass M slides down an identical inclined plane. No friction is acting on the block. If both objects are released at the same time, which one reaches the bottom first?
The block. The object with the largest ratio of translational kinetic energy to total kinetic energy is the one that wins the race. The block does not roll, so all of its kinetic energy is translational. The cylinder, however, rolls down the ramp, so its kinetic energy consists of a combination of rotational and translational kinetic energy. As a result, 100% of the block's kinetic energy is translational kinetic energy, while less than 100% of the cylinder's kinetic energy is translational. Since the block has a greater translational kinetic energy to total kinetic energy ratio, its center of mass will be moving faster than the cylinder's center of mass as the objects travel down the ramp. Therefore, the block will reach the bottom first.
400
A 90-kg man balances the boy on a teeter-totter as shown. The teeter-totter is uniform and has a mass of 20 kg. Goal: find the mass of the boy.
15 kg
400
A potter’s wheel is rotating with an angular velocity of ω = 3.2 rad/s. The potter applies a constant force, accelerating the wheel at 0.21 rad/s2. What is the wheel’s angular displacement after 6.4 s?
25 rad
400
A cricket batsman swings his bat, accelerating it uniformly from rest to 17.3 rad/s in 0.21 s. Assume the bat is modeled as a flat plate (I = 1/3 Mh2 + ½ Mw2), and has m = 1.36 kg, h=0.97 m, and w = 0.11 m. Find the torque applied by the batsman to the bat
36 N*m
400
A solid cylinder rotates with constant angular acceleration about a fixed axis. The cylinder’s moment of inertia (I) about the axis is 4.0 kg*m2. At time t = 0 s, the cylinder is at rest. At time, t = 2.0 s, its angular velocity is 4.0 rad/s. What is the rotational kinetic energy at t=2s?
32 J
500
A bullet of mass m=2.0g is moving horizontally with a speed of 500.0m/s. The bullet strikes and becomes embedded in the edge of a solid disk of mass M=3.2kg and radius R=0.5m. The cylinder is free to rotate around its axis and is initially at rest What is the angular velocity of the disk immediately after the bullet is embedded?
ωf=1.2rad/s.
500
A 90-kg man balances the boy on a teeter-totter as shown. The teeter-totter is uniform and has a mass of 20 kg. Determine the force, in newtons, that the fulcrum exerts on the teeter-totter. Hint: write the net force equation.
ΣFy=Fful−Wboy−WT−Wman=0
Fful=1225N
500
A Frisbee of radius 0.15 m is accelerating at a constant rate from 7.1 revolutions per second to 9.3 revolutions per second in 6.0 s. What is its angular acceleration?
2.3 rad/s2
500
A bowling ball of mass, M and radius, R, whose moment of inertia about its center is 2/5 MR^2, rolls without slipping along a level surface at a speed, v. It then encounters an incline and rolls up the incline. What is the maximum vertical height it will reach on the incline? Hint* Law of Conservation of Energy
7v2/10g
500
A 50-kg solid sphere rolls smoothly along a horizontal floor so that its center of mass has a speed of 5 m/s. What is the total kinetic energy (in joules) of the solid sphere? Note: you can look up the rotational inertia of a solid sphere.
K=(7/10)Mv2com=875JK