Please explain each step Consider a solid uniform ball of mass m and radius R rollingwithout slipping near the bottom of a frictionlesshemispherical fishbowl of radius r as shown. The balloscillates about the center of the fishbowl. Set thegravitatio nal potential energy to be zero at the bottom of thebowl.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 itsvelocity.Hint 1: Since the ball is rolling without slipping, its angular velocity can be expressed in terms of its linearvelocity, and it has translational and rotational energy.Hint 2: In writing gravitational potential U, use the approximation cos 0 ~ 1 –, then use SOH-CAH-TOAto 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.

Consider a solid uniform ball of mass m and radius R rolling without slipping near the bottom of a…

Answered: Consider a solid uniform ball of mass n…

Consider a solid uniform ball of mass n without slipping near the bottom hemispherical fishbowl of radius r oscillates about the center of the

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter7: Hamilton's Principle-lagrangian And Hamiltonian Dynamics

Section: Chapter Questions

Problem 7.5P

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Question

^{Please explain each step}

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.

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