A body weighing 62 newtons rests on a rough horizontal table such that the coefficient of friction between the body and the table is two-fifths. The body is connected by a light inextensible string passing over a smooth pulley fixed to the edge of the table to a weight of 24.8 newtons hanging freely vertically below the pulley. Is the body on the point of moving?
We will begin by sketching a diagram to model the situation. We are told that a body weighing 62 newtons rests on a rough horizontal table. There will be a normal reaction force 𝑅 acting vertically upwards. And as the table is rough, there will be a frictional force 𝐹 𝑟 acting against any motion. We are told that the coefficient of friction 𝜇 is equal to two-fifths. This body is connected to a weight of 24.8 newtons, which hangs freely vertically below a pulley. Since the pulley is smooth, there will be no friction and therefore the tension in the string will be equal throughout. As the string is light and inextensible, it has no mass and is a fixed length. And this also means that any acceleration will be constant for the whole system.
In this question, we are asked to work out whether the body is on the point of moving. In order to do this, we will resolve vertically and horizontally. If we consider the freely hanging body first, then as this body is at rest, we know that the sum of the forces equals zero. Letting the positive direction be vertically upwards, we have 𝑇 minus 24.8 equals zero. Adding 24.8 to both sides of this equation, we see that the tension in the string is equal to 24.8 newtons.
If we now consider the body resting on the table and once again resolve in the vertical direction where the sum of the forces equals zero, we have 𝑅 minus 62 is equal to zero. This time we can add 62 to both sides of our equation such that the normal reaction force 𝑅 is equal to 62 newtons. We know that when a body is on the point of moving, the friction force 𝐹 𝑟 is equal to 𝜇 multiplied by 𝑅, where 𝜇 is the coefficient of friction, in this case two-fifths.
The friction force 𝐹 𝑟 is therefore equal to two-fifths multiplied by 62, which is equal to 24.8. This friction force is equal to the tension in the string. And as these are the only two forces acting on the body on the table in the horizontal direction, we can conclude that yes, the body is on the point of moving. If the friction force was greater than the tension force, the body would not be on the point of moving. And if the tension force was greater than the friction force, the body on the table would be accelerating to the right and the freely hanging body would be accelerating downwards.