A body of mass 1.4 kilograms was placed on a smooth plane inclined 45 degrees to the horizontal. If a force of 59 newtons is acting on the body upwards along the line of greatest slope of the plane, determine the acceleration of the body rounded to the nearest two decimal places. Take 𝑔 to be equal to 9.8 meters per square second.
Let’s begin by sketching this out. Our plane is inclined at an angle of 45 degrees to the horizontal. A body of mass 1.4 kilograms is placed on this plane. That body exerts a downward force on the plane. The force is mass times acceleration due to gravity, so that’s 1.4𝑔. We are also told that a force of 59 newtons acts on the body upward along the line of greatest slope of the plane. In other words, that acts parallel to the plane. We’re going to use this information to calculate the acceleration.
Now, in fact, there is one further force, and that’s the normal reaction force. This force acts on the body by the plane and perpendicular to that plane. Since the plane is smooth, there are no frictional forces. In order to calculate the acceleration of the body, we’re going to use the formula 𝐅 equals 𝑚𝑎, force is mass times acceleration. And we’re going to use this formula in a direction that’s parallel to the plane. We know that we have a force of 59 newtons acting in this direction. But the weight of the body, that’s the downwards force that the body exerts on the plane, acts directly downwards.
And so we’re going to add this right-angled triangle to help us find the component of this force that acts parallel to the plane. We have an included angle of 45 degrees, and we’re looking to find this measurement. Let’s call that 𝑥 newtons. This represents the opposite side in this right-angled triangle, whereas our hypotenuse is 1.4𝑔. And so we can use the sine ratio, sin 𝜃 is opposite over hypotenuse. So we get sin of 45 is 𝑥 over 1.4𝑔.
If we multiply both sides of this equation by 1.4𝑔, we see that 𝑥 is 1.4𝑔 times sin of 45. And so we can now look at the forces that act parallel to the plane. We have 59 newtons acting up the plane. And then the component of the weight that acts parallel to the plane is acting in the opposite direction. So the resultant force on the body is 59 minus 1.4𝑔 times sin 45 degrees. This, of course, is equal to mass times acceleration.
Now we don’t know the acceleration, but we do know the mass is equal to 1.4. Evaluating the left-hand side, and we get 49.29 and so on. And we’re able to solve this equation by dividing through by 1.4. That gives us 35.213 and so on. Correct to two decimal places then, the acceleration of the body is 35.21 meters per square second.