2. A solid sphere of mass M and radius R rolls down an incline of inclination θ starting from a height h = 3 m. Your goal in this problem will be to find how fast the ball is moving at the bottom of the incline using two methods. The moment of inertia of a solid sphere rotating about its center of mass is 2/5 MR2 and the coefficient of static friction between the sphere and the incline is Us (static friction).
a. Use energy considerations to find an expression for the linear speed of the sphere at the bottom of the incline. You must include an energy bar chart in your solution for full credit.
b. Draw all the forces acting on the sphere while on the incline. (Hint: what must be the direction of the static friction force in order to create the required torque?)
c. Write down Newton’s 2nd law in component form for both the x and y directions.
d. Write down Tnet (net torque) = IA (inirtia * alpha) when the point of rotation is the center of the sphere.
e. Use the equations you wrote down in c and d to solve for the linear acceleration of the sphere. You will need to use the moment of inertia of a sphere and use the connection between linear and angular acceleration.
f. Use kinematics to find the speed of the sphere at the bottom of the incline.
g. Did you get the same answer as in a? If not, go back and check your work