Question

A ball of mass 2.5 2.5 kg and radius 0.145 0.145 m is released from rest on a plane inclined at an angle � = 35 θ=35 deg with respect to the horizontal. how fast is the ball moving (in m/s) after it has rolled a distance � = 1.95 d=1.95 m? assume that the ball rolls without slipping, and that its moment of inertia about its center of mass is 0.012 0.012 kg m2

Answer 1

To determine the ball's final speed after rolling down an incline, we use conservation of energy. The gravitational potential energy at the top is equated to the sum of translational and rotational kinetic energies at the bottom. The final speed is calculated by solving the energy conservation equation with the given mass, radius, moment of inertia, and the distance and angle of the incline.

Step-by-step explanation:

Calculating the Speed of a Ball Rolling Down an Incline

To calculate how fast a ball is moving after rolling down an inclined plane, we can apply the principle of conservation of energy.

The ball's gravitational potential energy at the top will be converted to kinetic energy and rotational kinetic energy as it rolls down.

Since the ball is rolling without slipping, we can use its translational kinetic energy and rotational kinetic energy to determine its final speed.

First, we calculate the potential energy at the top using the formula

PE = mgh,

where g is the acceleration due to gravity (9.81 m/s2), m is the mass of the ball, and h is the height of the incline which can be determined from the distance rolled and the angle of incline (sin(θ) = h/d).

Then we consider that the total energy at the top (potential energy) is equal to the total kinetic energy at the bottom,

which has two components - translational (KE_trans = 0.5 * m * v2) and rotational (KE_rot = 0.5 * I * ω2, where I is the moment of inertia and ω is the angular velocity).

The rolling condition relates these two via ω = v/r, where r is the radius of the ball.

Equating potential energy to the sum of translational and rotational kinetic energies, we can solve for the final velocity v with only one unknown.

This process gives us the final speed of the ball after rolling down the incline.

In this case, however, to provide a precise answer, we need the height of the incline, which is not provided in the problem statement.

Assuming the height can be determined from the given distance and angle, one could carry out the calculations as described.