A solid ball of radius 12.0 cm and mass 2.0 kg starts from rest (at position A i

Question

A solid ball of radius 12.0 cm and mass 2.0 kg starts from rest (at position A in the figure), rolls without slipping down the frictionless incline shown in the figure, and maximally compresses an initially unstrained spring by 5.0 cm. Take position A to be the reference at which the gravitational potential energy is 0. a. What is the total mechanical energy of the ball just before it makes contact with the spring (position B)? b. What is the speed of the ball just before it makes contact with the spring (position B)? c. What is the spring constant of the spring?

Explanation / Answer

a)

energy at B = PE + KE

EB = -m*g*L*sintheta + (1/2)*(7/5)*m*v^2

(b)

from energy conservation total mechanical energy is constant

EB = EA

-m*g*L*sintheta + (1/2)*(7/5)*m*v^2 = 0

m*g*L*sintheta = (1/2)*(7/5)*m*v^2

v = sqrt(10*g*L*sintheta/7)

v = sqrt(10*9.8*2.4*sin30/7)

v = 4.1 m/s <<<<------answer

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(c)

KE of the ball = PE stored in spring

(7/10)*m*v^2 = (1/2)*k*x^2

(7/10)*2*4.1^2 = (1/2)*k*0.05^2

k = 18827.2 N/m <<<------answer

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