Question Video: Finding the Acceleration of a Body Moving on a Smooth Plane Mathematics

Video Transcript

A body slides down a smooth plane under the action of its weight. Which one of the following variables does the acceleration of the body depend on? Is it (A) the reaction of the plane, (B) the weight of the body, (C) the mass of the body, or (D) the angle of inclination of the plane?

Let’s begin by sketching a diagram to model the scenario. If we let the mass of the body be 𝑚 kilograms, then its weight exerts a force vertically downwards equal to mass multiplied by gravity. The weight 𝑊 is equal to 𝑚𝑔. We have a normal reaction force 𝑅 acting perpendicular to the plane. And we will let the angle of inclination to the plane be 𝜃 degrees.

As the body is sliding down the plane under the action of its weight, there are no extra forces we need to consider. And as the plane is smooth, there will be no frictional force. We need to calculate the components of the weight force that are parallel and perpendicular to the plane. Using our knowledge of right angle trigonometry, the force perpendicular to the plane is equal to 𝑚𝑔 cos 𝜃 and the force parallel to the plane is equal to 𝑚𝑔 sin 𝜃.

Newton’s second law states that the sum of the forces is equal to the mass multiplied by acceleration. Resolving perpendicular to the plane, the sum of our forces is equal to 𝑅 minus 𝑚𝑔 multiplied by cos 𝜃. As the body is not accelerating in this direction, this is equal to zero. And the normal reaction force 𝑅 is therefore equal to 𝑚𝑔 multiplied by cos 𝜃. We can therefore conclude that the reaction of the plane does not impact the acceleration of the body.

Let’s now consider what happens when we resolve parallel to the plane. The only force acting on the body in this direction is 𝑚𝑔 multiplied by sin 𝜃. This is equal to the mass 𝑚 multiplied by the acceleration 𝑎. We can divide both sides of this equation by the mass 𝑚. This means that the acceleration 𝑎 is equal to 𝑔 sin 𝜃. And we can therefore conclude that neither the mass nor the weight of the body impacts the acceleration. The only variable that will affect the acceleration of the body is the angle of inclination 𝜃.

We can therefore conclude that the correct answer is option (D). The acceleration of the body depends on the angle of inclination of the plane. We can actually go one stage further. As 𝜃 lies between zero and 90 degrees, we know that as the angle 𝜃 increases, the acceleration increases. This is because the function sin 𝜃 is increasing between zero and 90 degrees.