A body is held at rest on a smooth plane, which is inclined to the horizontal at an angle 𝜃. The body is released and slides down the plane under the action of its weight. What is the acceleration of the body in terms of the gravitational acceleration 𝑔?
In this exercise, we want to solve for the acceleration that the body experiences. We’ll call that 𝑎. To start on our solution, let’s draw a sketch of the body at rest on the smooth plane. Here, we have a sketch of our body, which we’ve drawn as a box on an inclined plane at angle 𝜃. Our body has some mass. We can call that mass 𝑚.
Our first step with this diagram is to draw the forces that are acting on this body. We know that gravity acts on the body. We can call that force 𝐹 sub 𝑔. And there is also a normal force acting perpendicular to the inclined plane that acts on the body. We’ve called that force 𝐹 sub 𝑁. We’re told that the inclined plane is smooth. That’s a way of saying that there is no friction between the body and the plane surface. So there is no frictional force involved here.
Under the influence of these two forces, gravity and the normal force, the body will slide and accelerate down the plane. To find out what its acceleration is down the plane, we can set in place a pair of corded axes with 𝑦 pointing perpendicular to the inclined surface and 𝑥 pointing up the plane. Based on these two axes, we can separate the gravitational force 𝐹 sub 𝑔 into 𝑥- and 𝑦-components.
When we do that, we see that the 𝑦-component and the 𝑥-component form a right angle. So we have a right triangle, where those are the two legs and the magnitude of the gravitational force is the hypotenuse. It’s also true that the angle at the top corner of this right triangle is equal to the angle of our inclined plane 𝜃.
To solve for the acceleration of this body down the plane, we want to concentrate on the 𝑥-component of our gravity force. That particular component will be equal to the magnitude of the force of gravity 𝐹 sub 𝑔 multiplied by the sine of the angle 𝜃.
At this point, it will be helpful to recall Newton’s second law. The second law tells us that the net force acting on an object equals that object’s mass times its acceleration. In our example, if we focus solely in the 𝑥-direction, there is only one force at play. It’s the 𝑥-component of the gravitational force 𝐹 sub 𝑔 times the sine of 𝜃.
By the second law, we can say that this force is equal to the body’s mass times 𝑎 — its acceleration. We can rewrite 𝐹 sub 𝑔, which represents the gravitational force, as the product of the body’s mass times the acceleration due to gravity 𝑔. Written this way, we see that the mass of the body cancels out from both sides of the equation. Our result is independent of its mas. And this leads us to our answer for 𝑎. 𝑎 is equal to 𝑔 times the sine of 𝜃. That’s the acceleration of the body down the plane.