A solid sphere of 1.46kg and radius 0.1m, is rolling down a rough plane that is inclined at an angle 30 degrees to the horizontal. The 3 forces acting on the sphere are its weight, friction force and the normal reaction. The sphere starts rolling down when its point of contact with the plane is at the origin 0. By considering Newton's second law applied to the motion of the centre of mass of the sphere, calculate the magnitude of the normal reaction N, giving your answer to 3 decimal places.

Question

A solid sphere of 1.46kg and radius 0.1m, is rolling down a rough plane that is inclined at an angle 30 degrees to the horizontal. The 3 forces acting on the sphere are its weight, friction force and the normal reaction. The sphere starts rolling down when its point of contact with the plane is at the origin 0. By considering Newton's second law applied to the motion of the centre of mass of the sphere, calculate the magnitude of the normal reaction N, giving your answer to 3 decimal places.

Answer 1

To calculate the magnitude of the normal reaction force N, we need to consider the forces acting on the sphere.

1. Weight (mg): The weight of the sphere is given by the formula weight = mass × gravity. In this case, the mass of the sphere is 1.46 kg and the acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the weight of the sphere is 1.46 kg × 9.8 m/s^2 = 14.308 N.

2. Friction force (f): The friction force depends on the coefficient of friction (μ) and the normal reaction force (N). Since the sphere is rolling without slipping, we can use the relationship between the friction force and the normal reaction force given by f = μN. However, we don't know the coefficient of friction.

3. Normal reaction force (N): The normal reaction force is perpendicular to the plane surface and balances the component of the weight acting perpendicular to the plane. Since the plane is inclined at an angle of 30 degrees to the horizontal, the weight of the sphere can be split into two components: one parallel to the plane (mg sinθ) and the other perpendicular to the plane (mg cosθ), where θ is the angle of inclination.

Now, let's calculate the magnitude of the normal reaction force N:

First, find the component of weight parallel to the plane:
Weight_parallel = mg sinθ = 1.46 kg × 9.8 m/s^2 × sin(30°) ≈ 7.105 N.

Then, we can determine the component of weight perpendicular to the plane:
Weight_perpendicular = mg cosθ = 1.46 kg × 9.8 m/s^2 × cos(30°) ≈ 12.646 N.

Since the normal reaction force N balances the weight perpendicular to the plane, we find that N ≈ Weight_perpendicular = 12.646 N.

Therefore, the magnitude of the normal reaction force N is approximately 12.646 N.