To calculate the acceleration of a solid sphere rolling down an incline without slipping, we can consider two components: the linear acceleration (a) and the angular acceleration (α).

1. Linear acceleration (a):
The linear acceleration of the sphere can be determined using Newton's second law for rotation:
I * α = τ
where I is the moment of inertia of the sphere and τ is the torque acting on it.

For a solid sphere, the moment of inertia is given by:
I = (2/5) * m * R^2
where m is the mass of the sphere and R is the radius.

The torque acting on the sphere is due to the force of gravity pulling it down the incline. The component of gravity pulling the sphere down the incline can be decomposed into two forces:
- The force of gravity acting parallel to the incline (mg * sinθ), where θ is the angle of the incline.
- The normal force (mg * cosθ) acting perpendicular to the incline.

Since the sphere is in equilibrium, the torque acting on it is only due to the force of gravity acting parallel to the incline:
τ = (mg * sinθ) * R

Substituting the moment of inertia and torque into Newton's second law for rotation, we have:
((2/5) * m * R^2) * α = (mg * sinθ) * R

Simplifying the equation, we find:
α = (5/2) * (g/R) * sinθ

The linear acceleration of the sphere is equal to the linear acceleration of its center of mass:
a = α * R
= (5/2) * (g/R) * sinθ * R
= (5/2) * g * sinθ

2. Angular acceleration (α):
From the previous calculation, we found that α = (5/2) * (g/R) * sinθ.

3. Time to reach the bottom of the incline:
To determine the time it takes for the sphere to reach the bottom of the incline, we can use the following kinematic equation of motion:
v^2 = u^2 + 2 * a * s
where v is the final velocity, u is the initial velocity (which is 0 since the sphere starts from rest), a is the linear acceleration, and s is the distance traveled.

The distance traveled by the sphere down the incline can be calculated using the inclined plane formula:
s = L * cosθ
where L is the length of the incline.

Rearranging the kinematic equation, we have:
v = sqrt(2 * a * s)

Substituting the values of a and s, we get:
v = sqrt( 2 * (5/2) * g * sinθ * L * cosθ)

The time taken to reach the bottom of the incline is given by:
t = s/v

Substituting the value of s and v, we get:
t = (L * cosθ) / sqrt( 2 * (5/2) * g * sinθ * L * cosθ)

Simplifying the equation, we find:
t = sqrt( 2L / (5g) )

Therefore, the angular acceleration of the solid sphere rolling down the incline is (5/2) * (g/R) * sinθ, the linear acceleration is (5/2) * g * sinθ, and the time it takes to reach the bottom of the incline is sqrt( 2L / (5g) ).

Got a question with my answer?

Message Me

Cheapmath.com is completely free!

Asking question is free. Getting answers is free.

+99

Continue