Video Transcript
A body of mass 1.3 kilograms is moving with uniform velocity on a smooth plane inclined at 60 degrees to the horizontal. A force of 62 newtons is acting on the body along the line of greatest slope of the plane in an upward direction. Find the magnitude of the reaction of the plane. Take 𝑔 equal to 9.8 meters per second squared.
We will begin by sketching a diagram to model the scenario. We have a smooth plane inclined at 60 degrees to the horizontal. A body of mass 1.3 kilograms is moving with uniform velocity along the plane. We know that the body will exert a downward force equal to its weight. And this is equal to the mass of 1.3 kilograms multiplied by gravity, which in this question is 9.8 meters per second squared. There is a force acting vertically downwards equal to 1.3𝑔. Since the body is traveling with uniform velocity, its acceleration is equal to zero. We are also told that there is a force of 62 newtons acting along the slope of the plane in an upward direction.
Newton’s third law of motion tells us that there is a normal reaction force of the plane on the body. It is this reaction force 𝑅 that acts away from the plane at an angle of 90 degrees to the plane that we are trying to calculate. In order to do this, we will resolve forces perpendicular to the plane. And as a result, we firstly need to calculate the component of the weight force that acts in this direction.
Adding in a right triangle with included angle of 60 degrees, we can use our knowledge of right angle trigonometry. We are trying to calculate the length of the side adjacent to the angle. And as we know the hypotenuse, we will use the cosine ratio. This states that cos 𝜃 is equal to the adjacent over the hypotenuse. Substituting in the values we know, we have the cos of 60 degrees is equal to 𝑦 over 1.3𝑔. The cos of 60 degrees is one-half. And multiplying through by 1.3𝑔, we have 𝑦 is equal to 1.3𝑔 multiplied by one-half. This is equal to 6.37. The component of the weight that acts perpendicular to the plane is therefore equal to 6.37 newtons.
Since the body is moving with uniform velocity and, hence, the acceleration is zero, we know that the sum of the forces also equals zero. Taking the positive direction to be that of the reaction force, we have 𝑅 minus 6.37 is equal to zero. We can solve this equation by adding 6.37 to both sides such that the magnitude of the reaction of the plane is 6.37 newtons.
