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What is the final velocity of a hoop that rolls without slipping down a 5.00-m-high hill, starting from rest?
(a) Calculate the rotational kinetic energy of Earth on its axis. (b) What is the rotational kinetic energy of Earth in its orbit around the Sun?
(a) $2.57\times {\text{10}}^{\text{29}}\phantom{\rule{0.25em}{0ex}}\text{J}$
(b) ${\text{KE}}_{\text{rot}}=2\text{.}\text{65}\times {\text{10}}^{\text{33}}\phantom{\rule{0.25em}{0ex}}\text{J}$
Calculate the rotational kinetic energy in the motorcycle wheel ( [link] ) if its angular velocity is 120 rad/s. Assume M = 12.0 kg, R _{1} = 0.280 m, and R _{2} = 0.330 m.
A baseball pitcher throws the ball in a motion where there is rotation of the forearm about the elbow joint as well as other movements. If the linear velocity of the ball relative to the elbow joint is 20.0 m/s at a distance of 0.480 m from the joint and the moment of inertia of the forearm is $\text{0.500 kg}\cdot {\text{m}}^{2}$ , what is the rotational kinetic energy of the forearm?
While punting a football, a kicker rotates his leg about the hip joint. The moment of inertia of the leg is $\text{3.75 kg}\cdot {\text{m}}^{2}$ and its rotational kinetic energy is 175 J. (a) What is the angular velocity of the leg? (b) What is the velocity of tip of the punter’s shoe if it is 1.05 m from the hip joint? (c) Explain how the football can be given a velocity greater than the tip of the shoe (necessary for a decent kick distance).
A bus contains a 1500 kg flywheel (a disk that has a 0.600 m radius) and has a total mass of 10,000 kg. (a) Calculate the angular velocity the flywheel must have to contain enough energy to take the bus from rest to a speed of 20.0 m/s, assuming 90.0% of the rotational kinetic energy can be transformed into translational energy. (b) How high a hill can the bus climb with this stored energy and still have a speed of 3.00 m/s at the top of the hill? Explicitly show how you follow the steps in the Problem-Solving Strategy for Rotational Energy .
(a) $\text{128 rad/s}$
(b) $\text{19.9 m}$
A ball with an initial velocity of 8.00 m/s rolls up a hill without slipping. Treating the ball as a spherical shell, calculate the vertical height it reaches. (b) Repeat the calculation for the same ball if it slides up the hill without rolling.
While exercising in a fitness center, a man lies face down on a bench and lifts a weight with one lower leg by contacting the muscles in the back of the upper leg. (a) Find the angular acceleration produced given the mass lifted is 10.0 kg at a distance of 28.0 cm from the knee joint, the moment of inertia of the lower leg is $\text{0.900 kg}\cdot {\text{m}}^{2}$ , the muscle force is 1500 N, and its effective perpendicular lever arm is 3.00 cm. (b) How much work is done if the leg rotates through an angle of $\text{20.0\xba}$ with a constant force exerted by the muscle?
(a) $\text{10.}{\text{4 rad/s}}^{2}$
(b) $\text{net}\phantom{\rule{0.25em}{0ex}}W=6.\text{11 J}$
To develop muscle tone, a woman lifts a 2.00-kg weight held in her hand. She uses her biceps muscle to flex the lower arm through an angle of $\text{60.0\xba}$ . (a) What is the angular acceleration if the weight is 24.0 cm from the elbow joint, her forearm has a moment of inertia of $\text{0.250 kg}\cdot {\text{m}}^{2}$ , and the net force she exerts is 750 N at an effective perpendicular lever arm of 2.00 cm? (b) How much work does she do?
Consider two cylinders that start down identical inclines from rest except that one is frictionless. Thus one cylinder rolls without slipping, while the other slides frictionlessly without rolling. They both travel a short distance at the bottom and then start up another incline. (a) Show that they both reach the same height on the other incline, and that this height is equal to their original height. (b) Find the ratio of the time the rolling cylinder takes to reach the height on the second incline to the time the sliding cylinder takes to reach the height on the second incline. (c) Explain why the time for the rolling motion is greater than that for the sliding motion.
What is the moment of inertia of an object that rolls without slipping down a 2.00-m-high incline starting from rest, and has a final velocity of 6.00 m/s? Express the moment of inertia as a multiple of ${\mathit{MR}}^{2}$ , where $M$ is the mass of the object and $R$ is its radius.
Suppose a 200-kg motorcycle has two wheels like, the one described in Problem 10.15 and is heading toward a hill at a speed of 30.0 m/s. (a) How high can it coast up the hill, if you neglect friction? (b) How much energy is lost to friction if the motorcycle only gains an altitude of 35.0 m before coming to rest?
In softball, the pitcher throws with the arm fully extended (straight at the elbow). In a fast pitch the ball leaves the hand with a speed of 139 km/h. (a) Find the rotational kinetic energy of the pitcher’s arm given its moment of inertia is $\text{0.720 kg}\cdot {\text{m}}^{2}$ and the ball leaves the hand at a distance of 0.600 m from the pivot at the shoulder. (b) What force did the muscles exert to cause the arm to rotate if their effective perpendicular lever arm is 4.00 cm and the ball is 0.156 kg?
(a) 1.49 kJ
(b) $2.52\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N}$
Construct Your Own Problem
Consider the work done by a spinning skater pulling her arms in to increase her rate of spin. Construct a problem in which you calculate the work done with a “force multiplied by distance” calculation and compare it to the skater’s increase in kinetic energy.
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