# Rotational mechanics. cylinder over a plank

• vinay.ryder13
In summary, the question discusses a scenario where a solid cylinder and a plank are placed on a smooth horizontal surface. The plank is pulled with a horizontal force F, and the friction between the cylinder and the plank's surface prevents sliding. The acceleration of the center of the cylinder is being calculated using equations of motion and force. However, a small mistake has been made in the equations, leading to an incorrect solution. The correct equation for acceleration is acm = F/3m+M.

## Homework Statement

a solid cylinder of mass m and radius R and a plank of mass M are placed on a smooth horizontal surface. there is sufficient friction between the cylinder and the plank's surface to prevent sliding of the cylinder. the fixed pulley is smooth. if the plank is pulled with a horizontal force F, the acceleration of center of the cylinder is? (Refer the figure *dia.bmp*),(take tension in the string as T and friction between the cylinder & plank as f).

View attachment dia.bmp
2. The attempt at a solution

1. started off with the force equation of the plank => F- ( T + f)= Ma ----equation(a)

2. equation of motion of cylinder => T-f = macm -----equation(1)

3. rotational equation of cylinder => fR = Iα
fR = mR2/2 * α
f = macm/2

substituting in equation(1) and solving => T = 3macm/2

we know, acm = a/2 ( doubtful about the requisite)

so substituting the results in equation(a),

F - 2macm= 2Macm

=> acm= F/2(M+m).

but the solution given the referred book is acm = F/3m+M.

have i done some mistake is writing the equations or at some other point? please help! thanks.

Hello there ,
You are doing fine , just a small mistake is stopping you from getting the right answer .

The accelration of cylinder from grounds point of view would be twice of the accelration of block M.
T+f=2ma

and F+f-T=Ma
try to understand these and then solve again

@kushan - but, drawing fbd for plank is giving the eqn F-(f+T)=Ma. That is pricking me. Please can u verify it by an fbd? :)

If F-(f+T)=Ma is correct then the cylinder's eqn would also change.

Vinay ,
The friction is acting along the direction of F on the Cylinder
But [BOLD] according to Newtons law , friction in opp. direction should act on the slab [/BOLD]

Oh...i get it now. Thanks! :)

## 1. What is rotational mechanics?

Rotational mechanics is a branch of physics that deals with the motion of objects that rotate around a fixed axis. It involves the study of forces and torques that cause rotational motion, as well as the relationship between rotational and linear motion.

## 2. What is a cylinder over a plank?

A cylinder over a plank is a common example used in rotational mechanics to demonstrate the concepts of torque and rotational equilibrium. It refers to a cylinder placed on a flat surface, such as a wooden plank, and how it behaves when a force is applied to it.

## 3. What factors affect the rotational motion of a cylinder over a plank?

The rotational motion of a cylinder over a plank is affected by several factors, including the mass and shape of the cylinder, the length and angle of the plank, and the magnitude and direction of the applied force.

## 4. How is torque related to rotational motion?

Torque is the measure of the force that causes an object to rotate around an axis. In rotational mechanics, torque is directly related to the rotational motion of an object, as it determines the rate of change of an object's angular momentum.

## 5. What is the difference between rotational and linear motion?

Rotational motion refers to the movement of an object around a fixed axis, while linear motion refers to the movement of an object in a straight line. In rotational mechanics, the two types of motion are connected through the concept of torque, which can cause a change in both types of motion.