A wad of sticky clay with mass m = 2.15 kg and velocity v = 18.5 m/s is fired at

Question

A wad of sticky clay with mass m = 2.15 kg and velocity v = 18.5 m/s is fired at a wheel of moment of inertia 0.5 kg m2 and radius R = 0.541 m. The wheel is initially at rest and is mounted on a fixed horizontal axle that runs through its center of mass. The line of motion of the projectile is perpendicular to the axle and at distance d = 0.664R from the center.

Find the ratio Ef / Ei of the final to initial total energy of the system.

A wad of sticky clay with mass m = 2.15 kg and velocity v = 18.5 m/s is fired at a wheel of moment of inertia 0.5 kg m2 and radius R = 0.541 m. The wheel is initially at rest and is mounted on a fixed horizontal axle that runs through its center of mass. The line of motion of the projectile is perpendicular to the axle and at distance d = 0.664R from the center. Find the ratio Ef / Ei of the final to initial total energy of the system.

Explanation / Answer

According to conservation of angular momentum

L_i = L _ f

m v d = I w

Angular velocity

w = m v d / I

= (2.15 kg ) (18.5 m/s ) (0.664)(0.541 m ) / [ (0.5 kg m^2) + (2.15 kg)(0.541 m)^2 ]

= 12.65 rad/s

The ratio of final to the initial is

E f / Ei = ( 1/2 ) ( I w^2) / ( 1/ 2 (m v^2)

= [( 0.5 kg m^2) + (2.15 kg )(0.66)(0.541 m)] (12.65 rad/s) / (2.15 kg)(18.5 m/s)

=0.275

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