Question

A uniform cylinder of mass m, and radius r is pivoted on frictionless bearings. a massless string wrapped around the cylinder is connected to a block of mass me that is on a frictionless incline of angle 8 as shown below. the system is released from rest when the block is at a vertical distance h above the bottom of the incline. (use any variable or symbol stated above along with the following as necessary: g.)

Answer 1

To determine the acceleration of the block, use Newton's second law of motion. To find the mass of the hanging mass, use the equation for torque equilibrium. To find the wave speed, use the equation that relates tension, linear mass density, and mass.

Step-by-step explanation:

To determine the acceleration of the block, we can use Newton's second law of motion. The gravitational force acting on the block can be decomposed into two components: one parallel to the incline (mg*sin(45°)) and one perpendicular to the incline (mg*cos(45°)). The frictional force opposing the motion can be calculated by multiplying the coefficient of kinetic friction (0.4) by the normal force (mg*cos(45°)). Using these values, we can set up the equation:
a = (m*g*sin(45°) - µ*m*g*cos(45°))/(m + M), where m is the mass of the block and M is the mass of the pulley.

For the mass of the hanging mass (M), we can use the equation for torque equilibrium. The torque exerted by the gravitational force on the pulley is given by M*g*r, where r is the radius of the pulley. The torque exerted by the tension in the string is given by T*r, where T is the tension in the string. Since the system is in static equilibrium, the torques must cancel each other out, so we have:
M*g*r = T*r, which simplifies to M = T/g.

As for the wave speed of the wave traveling up the string, we can use the equation:
V = sqrt((T/µ)*(1/me)), where T is the tension in the string, µ is the linear mass density of the string, and me is the mass of the block on the incline.