Consider a solid uniform ball of mass m and radius R rolling without slipping near the bottom of a frictionless hemispherical fishbowl of radius r as shown. The ball oscillates about the center of the fishbowl. Set the gravitatio nal potential energy to be zero at the bottom of the bowl. rcos - con 8') Express the total energy of the ball as a function of the ball's distance from the center x and its A. velocity. Hint 1: Since the ball is rolling without slipping, its angular velocity can be expressed in terms of its linear velocity, and it has translational and rotational energy. Hint 2: In writing gravitational potential U, use the approximation cos 0 ~ 1-, thenuse SOH-CAH-TOA to rewrite 0 in terms of x and r, then use sin 0 0 to eliminate the variable 0.
Consider a solid uniform ball of mass m and radius R rolling without slipping near the bottom of a frictionless hemispherical fishbowl of radius r as shown. The ball oscillates about the center of the fishbowl. Set the gravitatio nal potential energy to be zero at the bottom of the bowl. rcos - con 8') Express the total energy of the ball as a function of the ball's distance from the center x and its A. velocity. Hint 1: Since the ball is rolling without slipping, its angular velocity can be expressed in terms of its linear velocity, and it has translational and rotational energy. Hint 2: In writing gravitational potential U, use the approximation cos 0 ~ 1-, thenuse SOH-CAH-TOA to rewrite 0 in terms of x and r, then use sin 0 0 to eliminate the variable 0.
Related questions
Question
100%
Transcribed Image Text:Consider a solid uniform ball of mass m and radius R rolling
without slipping near the bottom of a frictionless
hemispherical fishbowl of radius r as shown. The ball
oscillates about the center of the fishbowl. Set the
gravitatio nal potential energy to be zero at the bottom of the
bowl.
r(1- cos 0)
A.
Express the total energy of the ball as a function of the ball's distance from the center x and its
velocity.
Hint 1: Since the ball is rolling without slipping, its angular velocity can be expressed in terms of its linear
velocity, and it has translational and rotational energy.
Hint 2: In writing gravitational potential U, use the approximation cos 0 ~ 1 –, then use SOH-CAH-TOA
to rewrite 0 in terms of x and r, then use sin 0 ~ 0 to eliminate the variable 0.
`Find the oscillation frequency of the ball.
Hint: Since the ball is undergoing simple harmonic motion, x(t) (and hence v(t)) follows a certain form.
Also, since the energy of the system is conserved, then dE/dt = 0.
B.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
