A plate of mass m is placed on a frictionless surface. The plate is connected to block of mass M through a string over a massless pulley. A cylinder of mass m is placed on the plate which rolls without slipping. Find the frictional force acting on the cylinder:
Question
A plate of mass m is placed on a frictionless surface. The plate is connected to block of mass M through a string over a massless pulley. A cylinder of mass m is placed on the plate which rolls without slipping. Find the frictional force acting on the cylinder:
Solution
To solve this problem, we need to consider the forces acting on the cylinder and the plate separately.
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Forces on the cylinder:
- The gravitational force acting downwards is m*g.
- The normal force from the plate acting upwards is also m*g.
- The frictional force acting on the cylinder is f. This is the force we want to find.
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Forces on the plate:
- The gravitational force acting downwards is M*g.
- The normal force from the ground acting upwards is (M+m)*g.
- The tension in the string is T.
- The frictional force acting on the plate is -f (opposite to the friction on the cylinder).
Now, we can write down the equations of motion for the cylinder and the plate.
For the cylinder (using Newton's second law and the no-slip condition):
- m*a = f
- I*(alpha) = f*r
- where I is the moment of inertia of the cylinder, alpha is the angular acceleration, a is the linear acceleration and r is the radius of the cylinder. For a cylinder, I = 0.5mr^2 and alpha = a/r. Substituting these values in, we get f = 0.5ma.
For the plate (using Newton's second law):
- M*a = T - f
- Substituting f from the previous equation, we get Ma = T - 0.5m*a.
Solving these equations simultaneously, we can find the frictional force f.
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