Question

A uniform solid sphere rolls down an incline. A) What must be the incline angle (deg) if the linear acceleration of the center of the sphere is to have a magnitude of 0.21g? B) If a frictionless block were to slide down the incline at that angle, would its acceleration magnitude be more than, less than, or equal to 0.21g?

Answer 1

a) θ = 12.12°

b) equal to 0.21g

Step-by-step explanation:

Solution:-

Declare variables:

- The mass of solid sphere, m

- The inclination angle, θ

- The linear acceleration a down the slope of the solid sphere is a = 0.21g

Where, g: The gravitational acceleration constant.

- The component of weight of solid sphere directed down the slope is given by:

Ws = m*g*sin ( θ )

- Apply Newton's second law of motion down the slope, state:

F_net = m*a

- The only net force acting on the solid sphere is the Weight. So, the equation of motion in the coordinate axis ( down the slope ).

Ws = m*a

m*g*sin ( θ ) = m*0.21*g

- Solve for inclination angle ( θ ):

sin ( θ ) = 0.21

θ = arcsin ( 0.21 )

θ = 12.12°

- If a friction-less block of mass ( m ) moves down the same slope then block has weight component down the slope as:

Wb = m*g*sin ( θ )

- Apply Newton's second law of motion down the slope, state:

F_net = m*a

- The only net force acting on the solid sphere is the Weight. So, the equation of motion in the coordinate axis ( down the slope ).

Ws = m*a

m*g*sin ( θ ) = m*a

- Solve for linear acceleration ( a ):

g*sin ( θ ) = a

a = sin ( 12.12 ) * g

a = 0.21g

Answer: The acceleration is independent of mass and only depends on the inclination angle θ.