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 ferces 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.
To calculate the magnitude of the normal reaction, we need to consider the forces acting on the rolling sphere.
The forces acting on the sphere are its weight (mg), friction force (f), and normal reaction (N). Since the sphere is rolling without slipping, the friction force is static friction, which acts up the incline and opposes the motion of the sphere.
We can start by resolving the weight force and normal reaction force into their components. The weight force can be resolved into two components: one along the incline (mg sinθ) and one perpendicular to the incline (mg cosθ), where θ is the angle of inclination (30 degrees).
The normal reaction force can be resolved into two components: one perpendicular to the incline (N cosθ) and one along the incline (N sinθ).
Since the sphere is in rolling motion, we can write the equation for the net force acting on the sphere:
Net force = ma
The acceleration (a) of the sphere is related to the angular acceleration (α) of the rolling motion by the equation:
a = R * α
where R is the radius of the sphere.
Using Newton's second law for linear motion, we can relate the net force to the mass (m) and acceleration (a) of the center of mass of the sphere:
Net force = m * a
Substituting the expression for angular acceleration (α) in terms of the acceleration (a) and radius (R):
Net force = m * R * α
Now, let's substitute the force components into the equation:
mg sinθ - f = m * R * α
mg cosθ + N sinθ = m * a
N cosθ = mg
Since the sphere is rolling without slipping, the angular acceleration (α) can be related to the linear acceleration (a) by the equation:
α = a / R
Substituting this into the equation:
mg sinθ - f = m * R * (a / R)
mg cosθ + N sinθ = m * a
N cosθ = mg
Simplifying the equations:
mg sinθ - f = m * a
mg cosθ + N sinθ = m * a
N cosθ = mg
Now, we can solve these three equations simultaneously to find the value of the normal reaction force (N).
1. Start by substituting the expression for angular acceleration (α) into the first equation:
mg sinθ - f = m * R * (a / R)
mg sinθ - f = m * a
2. Substitute the expression for angular acceleration (α) into the second equation:
mg cosθ + N sinθ = m * a
3. Substitute the expression for normal reaction force (N) into the third equation:
(mg cosθ + N sinθ) * cosθ = mg
Now, let's solve the equations to find the value of N.
First, let's simplify the equations:
1. mg sinθ - f = m * a
2. mg cosθ + N sinθ = m * a
3. mg cosθ + N sinθ * cosθ = mg
Now, substitute the given values:
m = 1.46 kg
g = 9.8 m/s^2
θ = 30 degrees
R = 0.1 m
Substituting the values into the equations and solving simultaneously will give us the value of N.