Question

a 2.4 kg mass and a 3.8 kg mass are connected by a massless string over a pulley that is in the shape of a solid wheel having a radius 0.42 m and mass 4.9 kg. the two blocks hang vertically down. the pulley has a constant friction torque 5 nm. determine the linear acceleration of the blocks.

Answer 1

The linear acceleration of the blocks can be calculated by considering the gravitational forces, the tension in the string, and the constant friction torque on the pulley. Newton's second law and rotational dynamics are used, taking into account the moment of inertia and radius of the pulley.

Step-by-step explanation:

To determine the linear acceleration of the blocks connected through a massless string over a pulley with friction, we must consider both the forces acting on the blocks and the torque acting on the pulley. The forces on the two blocks include the gravitational forces (m1g and m2g), and the tension in the string. The torque on the pulley, due to its mass and the frictional force, affects how the system accelerates.

Let's denote the heavier mass as m2 (3.8 kg) and the lighter mass as m1 (2.4 kg). To calculate the acceleration (a), we apply Newton's second law and the rotational equivalent for the pulley. The net force for the system can be written as:

Fnet = m2g - m1g - ffriction, where ffriction is the friction force and can be calculated from the given torque (T) and radius (r) of the pulley as T = ffrictionr.

Since the pulley rotates as the blocks move, we consider angular acceleration (α) related to linear acceleration (a) by α = a/r. The moment of inertia (I) of the pulley (a solid wheel) is I = 0.5mpulleyr2. Using the torque equation T = Iα, we can solve for a. Finally, the linear acceleration of the blocks is found by dividing Fnet by the total mass (m1 + m2).