Multiple Choice
A hollow sphere of mass and radius of rolls without slipping down a high ramp angled at 20° above the horizontal. What is the speed of sphere when it reaches the bottom of the ramp? Ignore air resistance.
13. Rotational Inertia & Energy
Rotational Dynamics of Rolling Motion
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A hollow sphere 10 kg in mass and 2 m in radius rolls without slipping along a horizontal surface with 20 m/s. It then reaches an inclined plane that makes 37° with the horizontal, as shown. If it rolls up the incline without slipping, how long will it take to reach its maximum height? (Hint:You will need to first calculate its acceleration)
A
3.58 s
B
4.26 s
C
4.75 s
D
5.67 s
Verified step by step guidance1
Identify the forces acting on the sphere as it rolls up the incline: gravitational force, normal force, and frictional force. The frictional force provides the torque necessary for rolling without slipping.
Use the conservation of energy principle. Initially, the sphere has kinetic energy due to both its translational and rotational motion. As it moves up the incline, this energy is converted into gravitational potential energy.
Write the expression for the initial kinetic energy: \( KE_{initial} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \), where \( I \) is the moment of inertia of the hollow sphere \( I = \frac{2}{3}mr^2 \) and \( \omega = \frac{v}{r} \).
Set the initial kinetic energy equal to the gravitational potential energy at the maximum height: \( mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \). Solve for the height \( h \).
Calculate the acceleration of the sphere along the incline using \( a = g \sin(\theta) \), where \( \theta = 37^\circ \). Use the kinematic equation \( v_f = v_i + at \) to find the time \( t \) it takes to reach the maximum height, where \( v_f = 0 \).
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