Solving Friction Force Between Plank & Sphere

[ritwik06]

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

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

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?

 

[Doc Al]

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.

 

[thomate1]

I would first commend the student for their attempt at solving the problem and for providing a clear diagram and explanation of their thought process. I would then address their confusion about the negative sign in the friction force and explain that negative signs simply indicate the direction of the force, and in this case it is pointing in the opposite direction of the applied force. I would also suggest that the student double check their calculations and equations to ensure accuracy. Additionally, I would recommend using the correct units (e.g. N for force) and being consistent with the use of symbols (e.g. using a1 instead of a). Finally, I would encourage the student to think critically about the problem and to consider any assumptions that may have been made in their solution. Overall, the student has made a good effort in solving the problem and with some adjustments, they may arrive at the correct answer.