A body started moving from rest from the top of a ramp that is 312 centimeters long which was inclined at 60 degrees to the horizontal. When it reached the bottom, it continued moving on a horizontal plane. The resistance to the body’s motion is constant on both the ramp and the plane and equal to the square root of three divided by four times the weight of the body. Determine the distance that the body covered on the horizontal plane until it came to rest.
Let’s start by drawing a diagram. We’ll include the body, the ramp, and the horizontal plane at the bottom of the ramp. Here’s our diagram. We have the ramp labeled with its length of 312 centimeters, for which we’re going to use the letter 𝑙. We have the horizontal plane with the unknown distance represented by the letter 𝑑. And we have the body at its initial and final positions. Both the initial and final positions are labeled at rest because we are told that the body is at rest at both of these locations. Finally, we’ve also labeled the angle of 60 degrees that the ramp makes with the horizontal plane.
There are also two more quantities in the statement that we haven’t yet included in the diagram. They are the weight or force of gravity acting on the body and the resistance to the body’s motion. We’ll call the weight of the body 𝐹 sub 𝑊. And since weight is due to gravity, it always points in the downward direction. The resistance to the motion, on the other hand, always points exactly opposite the motion, so up the ramp when the object is moving down the ramp or backwards along the horizontal as the object is moving forwards. We’ll call the size of the resistance 𝐹 sub 𝑅. And as given in the question, this is the square root of three divided by four times the weight of the body.
Now that all of the information that we’re given is included in this diagram, we need a strategy for determining the unknown distance. The fact that we have information about the forces and distances in this problem suggests that we think about the quantity work which is related to forces acting on bodies as they move. Work is defined as the energy that an object gains or loses as it moves in the same or opposite direction to some external force. In particular, when the forces are constant and in the same direction as the motion, the work is exactly equal to the force applied times the distance the object moves.
In almost every instance in which we’re interested in using work, we are also interested in using the work–energy principle. The work–energy principle tells us that the net work, that is, the contribution from each individual force acting on an object between its beginning position and its final position, is exactly equal to the net changing kinetic energy between those two positions. With these two equations, we have found our strategy to answer this question. The change in kinetic energy is a quantity we can directly calculate knowing the body is initially and finally at rest. This change in kinetic energy is then exactly equal to the net work, which is the contribution from the weight and from the resistance to the motion.
Since work depends on distance, part of our calculation of the work from these two forces will include the unknown distance. So by calculating the kinetic energy and the work from each of the forces, we’ll get an equation that includes the single unknown, the distance, that we’re looking for. So let’s get started. Recall that kinetic energy is one-half times mass times speed squared. Now a body at rest, by definition, has a speed of zero, so its kinetic energy, one-half times mass times zero squared, is also zero. Since the final kinetic energy is zero and the initial kinetic energy is also zero, the total change in kinetic energy is zero.
Now let’s find the work from the weight and the resistance. We’ll start with the resistance. Since the resistance always points in the opposite direction to the motion, the motion and the force are parallel and we can use force times distance without modification. We know that the magnitude of the force is the square root of three divided by four times the weight of the body and the total distance traveled is 𝑙, the length of the ramp, plus 𝑑, the unknown distance along the horizontal plane. Note that we use the full distance 𝑙 plus 𝑑 because even though the direction of the resistance changes, it changes the same way that the motion changes. So the two are always parallel.
For the work done by the weight, we have to account for the fact that the weight points directly downward but the object either moves down the ramp or across the horizontal but never directly down. Recall that work is energy gained or spent moving in the same direction or opposite direction to a force. When the object is moving across the horizontal, it is moving perpendicularly to the downward direction. Since these two directions are perpendicular, they neither point in the same nor opposite direction. So the work done by the weight when the object is moving across the horizontal is zero.
On the other hand, when the body is moving down the slope, even though it’s moving at an angle, part of its motion is still in the downward direction. To find the work in this case, we need the distance that the body moved in the same direction as the force. The weight is directly downwards. In other words, it is purely vertical, so the distance that we’re looking for is just the vertical height of the body above the horizontal plane. Since this vertical line is perpendicular to the horizontal, we have a right triangle with a base angle of 60 and a hypotenuse of length 𝑙. Therefore the length of the vertical side is 𝑙 times the sin of 60 degrees or 𝑙 times the square root of three divided by two.
Okay, so the force acting on the body as it moves down the ramp is 𝐹 𝑊 and the total distance that the body travels in the same direction as 𝐹 𝑊 is 𝑙 times the square root of three divided by two. So the work done by the weight is 𝐹 sub 𝑊 times 𝑙 times the square root of three divided by two. Note that the horizontal distance associated with the ramp doesn’t affect the work done by the weight for the same reason the distance traveled along the horizontal plane doesn’t. In both cases because horizontal and vertical are perpendicular, then net contribution to the weight is zero.
Okay, now we have the net change in kinetic energy and the size of the work done by each of the forces. So we’re ready to combine them into the work energy principle. Since the resistance always points in the opposite direction to the motion, the proper combination for the net work is to subtract the work done by the resistance from the work done by the weight. This gives us, using the forms that we’ve already calculated, the work due to the weight minus the work due to the resistance equals zero.
Let’s now clear some space so we can solve this equation for the unknown 𝑑. Since this equation has zero on one side, if we multiply the other side by any nonzero quantity, the equation still holds. Since both terms on the left-hand side have an 𝐹 sub 𝑊 and a square root of three in them and neither 𝐹 sub 𝑊 nor square root of three is zero, let’s divide by these two quantities by multiplying by their reciprocals. While we’re at it, we can also get rid of the fractions by multiplying by four.
In the first term, 𝐹 𝑊 times the square root of three divided by the square root of three times 𝐹 𝑊 is just one and four divided by two is two. So this first term becomes two 𝑙. In the second term, four divided by four is one, square root of three divided by square root of three is one, and 𝐹 𝑊 divided by 𝐹 𝑊 is also one. This leaves us with 𝑙 plus 𝑑. And zero times anything is still zero. So now we have two 𝑙 minus 𝑙 plus 𝑑 equals zero. If we distribute the subtraction over the parentheses, we get two 𝑙 minus 𝑙 minus 𝑑 equals zero. Two 𝑙 minus 𝑙 is just 𝑙. And if we add 𝑑 to both sides, 𝑑 plus negative 𝑑 is zero and zero plus 𝑑 is 𝑑.
So we find that 𝑑, the distance along the horizontal plane that we’re looking for, is exactly equal to 𝑙 the length of the ramp. But we know that length. It’s given to us as 312 centimeters. So the answer that we’re looking for, the distance the body travels along the horizontal plane, is 312 centimeters.