Question Video: Analysis of the Motion of a Body on an Inclined Plane Mathematics • Third Year of Secondary School

Video Transcript

A body of mass 25 kilograms was placed on a smooth plane inclined at an angle 𝜃 to the horizontal. Sliding down the slope, the body traveled 20 meters in 10 seconds. After this, a force 𝐅 started to act on the body along the line of greatest slope up the plane. As a result of this force, the body began accelerating uniformly at 308 centimeters per square second up the slope. Determine sin 𝜃 and the force 𝐅. Take 𝑔 to be equal to 9.8 meters per square second.

Let’s begin by drawing a sketch of this scenario. The sketch doesn’t need to be to scale, but it should be roughly in proportion so we can accurately model what’s happening. Here is our smooth plane inclined at an angle 𝜃 to the horizontal. The fact that it’s smooth simply means that it won’t exert a frictional force on the body.

We’re told that a body with mass 25 kilograms sits on that plane. We can, therefore, say that it exerts a downward force on the plane. We call this downwards force its weight, and it’s equal to mass times acceleration due to gravity. For now, we’re going to call that 25𝑔 and that’s in newtons. We’re then told it slides down the slope. And in doing so, it travels 20 meters in 10 seconds. This means we’re able to model the acceleration of the body. Let’s begin by calculating that acceleration.

In the direction parallel to the plane, we know it travels a distance or displacement of 20, or 20 meters. It takes 𝑡 10 seconds to do that, and its initial velocity, 𝑣 naught, is zero. We’re looking to calculate the acceleration at this time. And so, we use one of our equations of constant acceleration. The one that we need that links 𝑠, 𝑡, 𝑣 naught, and 𝑎 is 𝑠 equals 𝑣 naught 𝑡 plus a half 𝑎𝑡 squared. Substituting what we know about the motion of this body into the equation and we get 20 equals 𝑎 plus a half times 𝑎 times 10 squared. This simplifies to 20 equals 50𝑎. And if we divide through by 50, we find 𝑎 is 20 over 50 or simply two-fifths.

And so, we’ve calculated the acceleration of the body. So, why is this useful? Well, it’s going to allow us to calculate the value of sin 𝜃. We’re going to use the equation 𝐅 equals 𝑚𝑎 and consider the first part of the motion of the body. This is force is mass times acceleration. We know the mass, and we’ve calculated the acceleration of the body. But the acceleration of this body acts parallel to the plane. So, we need to find the component of the weight that acts in this direction.

We, therefore, add a right-angled triangle. The included angle is 𝜃. And we want to find the opposite side in this triangle, the side that I’ve labeled 𝑥. Since we’re looking to find the opposite and we know the hypotenuse to be 25𝑔, we’re going to use the sine ratio. We can say that sin 𝜃 for this triangle is 𝑥 over 25𝑔, meaning 𝑥 equals 25𝑔 sin 𝜃. This is the component of the weight force that’s parallel to the plane. So, we can now substitute this into the formula 𝐅 equals 𝑚𝑎. We get 25𝑔 sin 𝜃 on the left-hand side. And on the right-hand side, we have mass times acceleration. So, that’s 25 times two-fifths. Now, 25 times two-fifths is 10.

So, our next job to calculate the value of sin 𝜃 is to divide through by 25𝑔. sin 𝜃 is 10 divided by 25𝑔. And now, we use the fact that 𝑔 is 9.8 meters per square second. And we find sin 𝜃 is 10 divided by 25 times 9.8, which gives us two over 49. And so, we’ve answered the first part of this question. We’ve determined the value of sin 𝜃 to be two over 49.

Next, we need to find the force 𝐅. So, we’re going to need to redraw our diagram to model the second part of the motion. The weight force remains unchanged, and in fact we can change the opposite side in the triangle that we drew to be 25𝑔 sin 𝜃. This will be useful going forward. We now know that we can calculate the component of the weight that’s parallel to the plane.

We then have this force 𝐅 that acts on the body along the line of greatest slope up the plane, so it’s parallel to the plane. There is also a normal reaction force of the plane on the body, but we don’t really need that in this question. We’re told that the body accelerates uniformly at 308 centimeters per square second. Now, we should convert this to meters per square second. And to do so, we divide by 100. And we find the acceleration is 3.08 meters per square second.

We’re looking to find the value of the force 𝐅, so we’re going to go back to our equation 𝐅 equals 𝑚𝑎. We know the mass and the acceleration of the body, but what is the force this time? Well, we need to consider the sum of the forces. Let’s take the positive direction to be now going up the plane, and we have 𝐅 acting in this direction. Then acting in the opposite direction, we have 25𝑔 sin 𝜃. So, the sum of the forces acting on the body now — assuming up the plane to be positive — are 𝐅 minus 25𝑔 sin 𝜃. This is equal to mass times acceleration, so it’s equal to 25 times 3.08.

Let’s replace sin 𝜃 with two over 49 and we know that 25 times 3.08 is 77. Then, if we replace 𝑔 with 9.8, this becomes 25 times 9.8 times two over 49, which is 10. And our equation becomes 𝐅 minus 10 equals 77, which we can solve for 𝐅 by adding 10 to both sides. And when we do, we get 𝐅 equals 87 or 87 newtons. So, sin 𝜃 is two over 49, and the force 𝐅 is 87 newtons.