A hollow spherical shell with mass 2.45kg rolls without slipping down a slope that makes an angle of 30.0∘ with the horizontal. a.) find the magnitude of the acceleration of the center of mass of the spherical shell. b.) find the magnitude of the frictional force acting on the spherical shell.

I know that the center of mass for a hollow sphere is 2/5mr^2 and since the sphere is not slipping v=wR (w=omega) and a=(alpha)*r but I'm not sure how to approach this question.

To solve this problem, you can use the equations of rotational and translational motion. The key is to relate the angular acceleration of the sphere to its linear acceleration.

a) To find the magnitude of the acceleration of the center of mass of the spherical shell, you can use the equations of rotational motion. The linear acceleration of the center of mass is equal to the product of the angular acceleration and the radius of the sphere.

The net torque acting on the sphere is due to the force of gravitational torque and the frictional torque. Since the sphere is rolling without slipping, the frictional torque is equal to the gravitational torque. The gravitational torque is given by the product of the weight of the sphere and the radius of the sphere.

The torque equation is given by:

Net Torque = Moment of Inertia * Angular Acceleration

The moment of inertia of a hollow sphere is (2/3) * mass * radius^2.

The net torque acting on the sphere is the difference between the magnitudes of the gravitational torque and the frictional torque.

Gravitational Torque = mg * r * sin(θ), where m is the mass of the sphere, g is the acceleration due to gravity, r is the radius of the sphere, and θ is the angle of the slope with the horizontal.

The frictional torque is μ * N * r, where μ is the coefficient of friction and N is the normal force. Since the sphere is rolling without slipping, the normal force is equal to the weight of the sphere.

Therefore, the equation for the net torque is:

(2/3) * m * r^2 * α = mg * r * sin(θ) - μ * mg * r,

where α is the angular acceleration. We need to find the linear acceleration, a, which is equal to α * r. Rearranging the equation, we get:

a = (3/2) * (g * sin(θ) - μ * g).

Substituting the given values, we have:

a = (3/2) * (9.8 m/s^2 * sin(30°) - μ * 9.8 m/s^2).

Calculate the value of a using the above equation.

b) To find the magnitude of the frictional force acting on the spherical shell, you can use the equation for the frictional force. The frictional force is equal to the coefficient of friction times the normal force.

The normal force is equal to the weight of the sphere, which is mg.

Therefore, the equation for the frictional force is:

Frictional Force = μ * mg.

Substitute the value of μ and mg in the equation to find the magnitude of the frictional force.

To solve this problem, we can use the principles of rotational motion and Newton's second law. Let's break it down step by step:

a.) To find the magnitude of the acceleration of the center of mass of the spherical shell, we need to consider the forces and torques acting on it.

1. Forces acting on the spherical shell:
- Gravitational force (mg): acts vertically downwards
- Normal force (N): acts perpendicular to the incline
- Frictional force (f): acts in the opposite direction of motion

2. Torques acting on the spherical shell:
- Gravitational torque: since the mass is symmetrically distributed, the torque is negligible.

Since the spherical shell is rolling without slipping, the frictional force provides the necessary torque for rotation. The frictional force can be expressed as f = Iα, where I is the moment of inertia and α is the angular acceleration.

The moment of inertia for a hollow sphere is I = (2/3)mr^2, where m is the mass of the shell and r is the radius.

Now, we can write the equations of motion:
- ΣFy = N - mgcosθ = 0 (vertical equilibrium)
- ΣFx = f - mgsinθ = ma (horizontal acceleration)

Substituting the expression for the moment of inertia into the equation f = Iα:
- f = (2/3)mr^2 * α

Since the angular acceleration α is related to the linear acceleration a by α = a/r, we can rewrite the equation as:
- f = (2/3)ma * r (equation 1)

To find the magnitude of the acceleration a:
- Solve the equation of motion ΣFx for a:
a = (f - mgsinθ) / m

- Substitute equation 1 into the equation for a:
a = (2/3)g * r - gsinθ

b.) To find the magnitude of the frictional force acting on the spherical shell, we can use the equation for the frictional force:

- f = (2/3)ma * r

Now, plug in the known values:
- m = 2.45 kg (mass of the shell)
- r = radius of the shell
- g = acceleration due to gravity (9.8 m/s^2)
- θ = 30 degrees

Calculate the values of a and f using the above equations, substituting the known values and solving for the unknowns.

can i get the answer