Using the conservation of mechanical energy, calculate the final speed and the acceleration of a cylindrical object of mass m and radius r after it rolls a distance s without slipping along an inclined plane that makes an angle β with respect to the horizontal.

speed is 4/3 × g × sin β

acceleration is g sin θ

Explanation:

Given data

mass = M

radius = R

distance = s

angle = β

to find out

final speed and the acceleration

solution

we know that cylinder is rolling on inclined plane

so from conservation of mechanical energy

energy at top = energy at bottom

energy at top will be

= m ×g×h

= m ×g×s sinβ    1

and

energy at bottom will be

= 1/2 ×m × v² + 1/2 × z × ω²

= 1/2 ×m × v² + 1/2 × 1/2 ×m× r² × (v/r)²

= 3/4 × v²    2

equating now equation 1 and 2

m ×g×s sinβ = 3/4 × v²  

so v = 4/3 × g × sin β

so speed is 4/3 × g × sin β

and acceleration is

= mg sin θ    3

and

acceleration = ma   4

now equation equation 3 and 4

ma = mg sin θ

so acceleration is g sin θ

c. magnitude and direction