The pulley has an inner radius of 0.35m and an outer radius of 0.65m. It has a mass of 1.8kg. A string wrapped around the inner part of the pulley is fastened to the ceiling. A second string wrapped around the outer part of the pulley is fastened to a block. Determine the mass of the block if the pulley remains at rest.
Please do show you did it and which formula did you use. Thanks very much.
The string attached to the ceiling supports the pulley and the block.
The tension of this string is T = (1.8 + m) •g,
the torque of this force is
T•r =(1.8 + m) •g•r.
The torque created by the block
is m•g•R .
The condition for equilibrium:
the net torque is zero.
(1.8 + m) •g•r = m•g•R,
m = 1.8•r/(R-r) =
= 1.8•0.35/(0.65 – 0.35) = 2.1 kg
To determine the mass of the block, we need to analyze the forces acting on the system.
First, let's consider the forces acting on the pulley. The only force acting on the pulley is the tension in the string that is fastened to the ceiling. This tension force will produce a torque on the pulley, causing it to rotate.
Next, let's consider the forces acting on the block. The weight of the block (mg) will act vertically downwards and must be balanced by the tension in the string wrapped around the outer part of the pulley. This tension force will prevent the block from falling.
Since the pulley is at rest, the net torque acting on it must be zero. This means the torque caused by the tension in the string wrapped around the inner part of the pulley (τ1) must be equal in magnitude and opposite in direction to the torque caused by the tension in the string wrapped around the outer part of the pulley (τ2).
The torque caused by the tension in the string wrapped around the inner part of the pulley can be calculated using the equation: τ1 = I1 * α, where I1 is the moment of inertia of the pulley with respect to the axis of rotation (given by ½ * m * r1^2, where m is the mass of the pulley and r1 is the inner radius of the pulley) and α is the angular acceleration of the pulley (which is zero as the pulley is at rest).
Similarly, the torque caused by the tension in the string wrapped around the outer part of the pulley can be calculated using the equation: τ2 = I2 * α, where I2 is the moment of inertia of the pulley with respect to the axis of rotation (given by ½ * m * r2^2, where r2 is the outer radius of the pulley).
Since τ1 is equal in magnitude and opposite in direction to τ2, we can equate the two equations to eliminate α:
I1 * α = I2 * α
Substituting the expressions for I1 and I2 and rearranging the equation, we get:
½ * m * r1^2 * α = ½ * m * r2^2 * α
Canceling out the common terms and rearranging the equation, we get:
r1^2 = r2^2
Substituting the given values, we have:
(0.35)^2 = (0.65)^2
Simplifying the equation, we get:
0.1225 = 0.4225
Since the equation is not true, it means that our assumption that the block is at rest is incorrect. Therefore, we cannot determine the mass of the block with the given information.