The acceleration of a ball rolling down an incline can be determined using the principles of rotational motion and the effects of gravity. When a ball rolls down an incline, two types of forces contribute to its acceleration:
Gravitational force: The force of gravity acts vertically downward on the ball. This force can be resolved into two components: one parallel to the incline (mg sinθ) and one perpendicular to the incline (mg cosθ), where m is the mass of the ball and θ is the angle of the incline.
Rolling resistance: When a ball rolls, there is a rolling resistance that opposes its motion. This resistance arises due to the interaction between the ball and the surface on which it is rolling. Rolling resistance depends on factors such as the nature of the ball's surface and the presence of any external factors like friction.
Assuming the rolling resistance is negligible or taken into account separately, we can focus on the component of gravitational force parallel to the incline (mg sinθ). This component accelerates the ball down the incline.
The acceleration (a) of the ball rolling down the incline can be calculated using Newton's second law, considering the net force acting on the ball:
Force = mass × acceleration
mg sinθ = m × a
Simplifying the equation, we find:
a = g sinθ
Therefore, the acceleration of a ball rolling down an incline is equal to the acceleration due to gravity (g) multiplied by the sine of the angle of the incline (θ). This equation holds true under the assumption that the rolling resistance is negligible or separately accounted for.