# A plank and sphere

## Homework Statement

A plank of mass m is placed on a smooth surface. Now a uniform solid sphere of mass m and radius R is placed on the plank as shown in the figure. A force F is applied at top most point of the sphere at an angle of 45 to the horizontal. Surface between the plank and the sphere is extremely rough so that there is no slipping. Find the force of friction acting between the plank and the sphere.

http://img76.imageshack.us/img76/7259/diagin7.jpg [Broken]

## The Attempt at a Solution

This is the diagram I drew:
I considered only the necessary forces. Normals have been omitted.
http://img82.imageshack.us/img82/3304/freebodydiagramoa9.jpg [Broken]

By torque equation:
$$(F/\sqrt{2}+Fr)*R=2/5 MR^{2}* a/R$$

I can write angular acceleration with respect to plank as a/R, since the boy does not skid.
$$(F/\sqrt{2}+Fr)=ma1$$
a1 is acceleration with respect to ground.

a1=a- Fr/m

I solve these 3 equations and I got$$Fr= -3F/\sqrt{2}$$
but the correct answer given seems like $$Fr= -F/3\sqrt{2}$$
An I also have one more confusion: Why is friction force coming out to be negative???

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## Answers and Replies

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By torque equation:
$$(F/\sqrt{2}+Fr)*R=2/5 MR^{2}* a/R$$

I can write angular acceleration with respect to plank as a/R, since the boy does not skid.
$$(F/\sqrt{2}+Fr)=ma1$$
a1 is acceleration with respect to ground.
These equations aren't consistent. If, as you assumed, the friction on the sphere points to the right, then it exerts a torque opposite to that of the applied force.

An I also have one more confusion: Why is friction force coming out to be negative???
The sphere drags the plank to the right, thus the friction force on the sphere points left.