Flying Circus of Physics A bowler throws a bowling ball of radius R = 11 cm alon

Question

Flying Circus of Physics A bowler throws a bowling ball of radius R = 11 cm along a lane. The ball (the figure) slides on the lane with initial speed Vcom, 0 = 8.0 m/s and initial angular speed w0 = 0. The coefficient of kinetic friction between the ball and the lane is 0.26. The kinetic frictional force Fk acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When speed Vcom has decreased enough and angular speed psi has increased enough, the ball stops sliding and then rolls smoothly. During the sliding, what are the ball?s (a) linear acceleration and (b) angular acceleration? (c) How long does the ball slide? (d) How far does the ball slide? (e) What is the linear speed of the ball when smooth rolling begins? Note that the clockwise direction is taken as negative.

Explanation / Answer

a)

Force of friction, f = umg

where u = coefficient of friction = 0.26

g = 9.8 m/s2

So, acceleration, a = f/m = ug

So, a = 0.26*9.8 = 2.55 m/s2 <------- linear acceleration(a)

b)

Torque , T = f*R

where R = radius of ball = 0.11 m

Now, torque , T = I*A

where I = moment of inertia = (2/5)*mR^2

A = angular acceleration

So, T = fR = (2/5)*mR^2*A

So, f = (2/5)*mR*A

So, umg = (2/5)mR*A

So, A = (5/2)*ug/R

So, A = (5/2)*0.26*9.8/0.11 = 57.9 rad/s2 <-----------answer

c)

Using the equation of motion.

W = Wo + A*t

where W = angular speed at time t

Wo = inial angular speed = 0

So, W = A*t <------------ eq(1)

For linear motion, using the equation, v = u +at

u = vcom,0 = 8 m/s

v = 8 + (-2.55)*t <---- NOTE : the -ve sign for friction acting opposite to motion

For rolling motion, v = W*R

So, 8 - 2.55*t = A*R*t

So, 8 - 2.55*t = 57.9*0.11*t

So, t = 0.897 s <---------- answer

d)

Using the equation of motion,

s = ut + 0.5*at^2

So, s = 8*0.897 - 0.5*2.55*0.897^2 = 6.15 m <-------answer

e)

linear speed when rolling begins, v = u + at

= 8 - 2.55*0.897

= 5.7 m/s <----------answer

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