A uniform solid sphere rolls down an incline without slipping. If the linear acceleration of the center of mass of the sphere is 0.21g, then what is the angle the incline makes with the horizontal?
To find the angle the incline makes with the horizontal, we can use the concept of torque and kinematics. Here's how you can solve it step by step:
1. Start by drawing a free body diagram of the rolling sphere. We have the weight of the sphere (mg) acting downward, and the normal force (N) and frictional force (f) acting upward and parallel to the incline.
2. Identify the forces acting on the sphere. The weight of the sphere can be broken down into two components: mg * cosθ acting parallel to the incline, and mg * sinθ acting perpendicular to the incline.
3. Apply Newton's second law in the direction parallel to the incline. The net force in this direction is given by: f - mg * sinθ = ma, where f is the frictional force and a is the linear acceleration of the center of mass.
4. Apply Newton's second law in the direction perpendicular to the incline. The net force in this direction is given by: N - mg * cosθ = 0, since there is no acceleration in this direction.
5. Determine the relationship between the frictional force and the normal force. The frictional force is given by f = μN, where μ is the coefficient of friction between the sphere and the incline. Substitute this expression into the parallel equation from step 3.
6. Substitute the expression for the normal force from step 5 into the perpendicular equation from step 4.
7. Solve the resulting equations to find the value of the angle θ.
8. Substitute the given value of the linear acceleration (0.21g) into the parallel equation and solve for θ.
By following these steps, you can find the angle the incline makes with the horizontal.