Velocity and Acceleration in Experimental Setup

What is the linear acceleration of the mass after it has fallen a distance x?

Answer symbolically. HINT: Sketch the situation and construct a free-body diagram.

What is the angular velocity of the disc at the same moment?

Consider kinematic relationships and the connection between linear and angular quantities when answering this question. Answer symbolically.

The linear acceleration of the mass after it has fallen a distance x is (mgr^2)/(I + mr^2), while the angular velocity of the disc at the same moment is (mgt)/(I/r + mr).

The linear acceleration of the mass after it has fallen a distance x can be found by using the equations of motion for the mass and the disc. The free-body diagram for the mass shows that the only forces acting on it are the force of gravity (mg) and the tension in the string (T). The free-body diagram for the disc shows that the only torque acting on it is due to the tension in the string (Tr).

Using Newton's second law for the mass, we get: mg - T = ma. Where a is the linear acceleration of the mass. Using Newton's second law for the disc, we get: Tr = Iα. Where I is the moment of inertia of the disc and α is the angular acceleration of the disc.

Since the string is wrapped around the pulley, the linear acceleration of the mass is related to the angular acceleration of the disc by the equation: a = αr. Substituting this into the equation for the torque on the disc, we get: Tr = I(a/r). Solving for the tension in the string, we get: T = (Ia)/(r^2).

Substituting this back into the equation for the forces on the mass, we get: mg - (Ia)/(r^2) = ma. Rearranging and solving for the linear acceleration, we get: a = (mgr^2)/(I + mr^2). This is the linear acceleration of the mass after it has fallen a distance x.

The angular velocity of the disc at the same moment can be found by using the kinematic relationship between linear and angular quantities: v = ωr. Where v is the linear velocity of the mass and ω is the angular velocity of the disc. Since the linear acceleration of the mass is related to the angular acceleration of the disc by the equation: a = αr.

We can use the equations of motion to relate the linear velocity of the mass to the linear acceleration: v = v0 + at. Since the mass starts from rest, v0 = 0, and we can substitute the equation for the linear acceleration into this equation to get: v = (mgr^2t)/(I + mr^2). Substituting this back into the equation for the angular velocity, we get: ω = (v/r) = (mgr^2t)/(r(I + mr^2)).

Simplifying, we get: ω = (mgt)/(I/r + mr). This is the angular velocity of the disc at the same moment.

Understanding the relationship between linear acceleration and angular velocity in this experimental setup helps confirm the expression derived using energy conservation arguments. By applying kinematics and dynamics principles, we can further validate the results obtained. This process of analyzing forces, torques, and motion equations provides a comprehensive understanding of the system's behavior.