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## Homework Statement

A bowling ball of mass m and radius R sits on the smooth floor of a subway car. If the car has a horizontal acceleration ##\mathbb{a_1}## , what is the acceleration ##\mathbb{a_2}## of the ball? Assume that the ball rolls without slipping. The gravitational acceleration is g.

## Homework Equations

c is the center of mass ( just a reference point)

Rolling motion: ##a_c## = ##α_cR##

Torque : ##τ_c## = ##I_cα_c##

Newton's 2nd Law: ##F## = ##ma##

Moment of Inertia of a sphere rotating about an axis through its center: ##\frac {2MR^2} {5} ##

## The Attempt at a Solution

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Some predictions I had (may or may not be right):

Smooth floor; therefore there is no static friction between the floor and the ball.

Yet the ball is rolling; therefore there must be a horizontal force F acting on the ball ( I call it a "generated" friction force produced somehow by the acceleration of the subway car floor. Not sure how this is produced, though.) The rolling motion could not have been produced by the vertical forces, for the only vertical forces are weight and the contact force between the ball and the floor. The contact force must always be acting along the same line of action as the center of mass c, through which gravity acts, so there is no possibility of either force producing a torque with respect to the center of mass.

The only possible force, is a horizontal force F that acts at the contact between the floor and the ball.

This force is the one responsible for the acceleration ##a_2## experienced by the ball, so $$F = ma_2 (1)$$

This force is also responsible for the torque about c, so $$τ_c = FR = I_cα_c (2)$$

The ball is rolling without slipping, so $$ a_2 = α_cR (3)$$

And this is where I am stuck, because manipulating equations (1) and (2) and making ##a_2## the subject does not give me equation (3). So one of my equations must be wrong, and yet they seem physically sound and valid to me.

Can anyone out there provide an insight? Really sorry if this seems trivial and the answer is actually staring at me.