Video Transcript
A ball accelerates as it rolls down a slope, continues to roll along a horizontal surface at constant speed, and then again accelerates as it rolls down a second slope. Which of the following correctly shows the forces acting on the ball at equal time intervals?
We can see that we’ve been given three different diagrams labeled as (A), (B), and (C). Each of these diagrams shows the position of a ball at equally spaced intervals of time. There are three different sections to the ball’s motion, and this is seen most clearly by looking at the diagram given in option (C). There’s this first bit here where the ball is rolling down a slope. And we can see that during this part of the motion at equally spaced time intervals when the ball’s position is shown, the distance traveled by the ball increases with each subsequent interval. That’s because during this part of the motion, the ball’s speed is increasing as it accelerates down the slope.
We then have this second part of the motion where the ball is rolling along a horizontal surface. We’re told that at this point the ball is moving at a constant speed. And indeed, if we look at the equally spaced time intervals at which the ball’s position is shown, we can see that the distance moved by the ball in each of these time intervals is the same. So the ball is moving equal distances in equal times. Lastly, we have this second downhill segment. And here once again, the ball is accelerating. We can notice that the positions the ball is shown at at these equal time intervals is the same in all of the diagrams (A), (B), and (C) that we’re given. The thing that differs between the three diagrams is the arrows drawn on the ball at each point representing the forces acting on the ball at that instant in time.
We’re being asked to work out which of these three diagrams correctly shows these forces. To work this out, it’s going to be helpful to recall a law known as Newton’s first law of motion. This law says that an object at rest will remain at rest and an object moving with a constant velocity will continue at that velocity unless acted on by an unbalanced force. What this law is telling us is that if there is no unbalanced force acting on an object, that is, if the forces on it are balanced or equivalently there is no net force, then the velocity of that object won’t change. That means that if the object isn’t initially moving, so it’s initially at rest, then it will remain at rest. Meanwhile, if we’ve got a moving object with no net force on it, then that object will continue to move at the same constant velocity.
In this question, we’re considering a ball that’s moving. So it’s the second part of Newton’s first law about moving objects that we’re interested in. We can summarize this part of Newton’s first law as saying that if there is no net force on a moving object, then that object continues moving at a constant speed. Now, if this is the case, then the argument must also work the other way round. That is, if we have an object that’s moving at a constant speed, we know that the net force on that object must be zero. Thinking about the ball from this question, we know that during the second part of the motion, when it’s rolling along a horizontal surface, the ball is moving at a constant speed.
If we look at the diagram given in option (A), we can see that during this section of motion, there is a force vector on the ball at each of the time intervals shown. Notice that there’s just the one arrow drawn on the ball in each case pointing in the direction of the ball’s motion. This arrow represents a single unbalanced force acting on the ball in this direction. That is, since there are no other forces shown to balance this force out, then according to this diagram shown in option (A), during this section of motion, there is a net force on the ball in the direction of its motion.
However, we’ve seen from Newton’s first law of motion that if an object is moving at a constant speed, then the net force on it must be zero. And we also know because we’re told in the question text that during this section of motion on the horizontal surface, the speed of the ball is constant. That means that we know that the net force on the ball must be zero during this horizontal part of its motion. Since that isn’t the case in the diagram shown in option (A) and we do have a net force on the ball at this point in the motion, then we can safely eliminate this answer choice. This leaves us with the diagrams shown in options (B) and (C).
In both cases, during the horizontal part of the ball’s motion, we can see that there’s a net force of zero acting on the ball. In diagram (B), the net force is zero because there are no forces drawn at all when the ball is moving on a horizontal surface. Meanwhile, in diagram (C), there are two forces indicated at each point in the ball’s motion when it’s on the horizontal surface. At each point, we see a downward force indicated in red and an upward force indicated in blue.
Let’s recall that when we represent a force by an arrow, the length of that arrow gives us the magnitude of the force. We can notice that when the ball is on the horizontal, the red arrow and the blue arrow have the same length as each other. This means that these two forces are equal in magnitude, and we can also see that they are oppositely directed. We can say that these forces are balanced because they cancel each other out such that there is no net or resultant force on the ball at this point. What we’ve seen so far then is that as far as Newton’s first law of motion is concerned, either diagram (B) or diagram (C) could be correct. This is true because in both cases, when the ball is moving at a constant speed, these diagrams show a net force on the ball of zero.
Newton’s first law then can’t help us to distinguish between these two diagrams. But there’s also another consideration we can use to help us, and that’s the consideration of gravity. The gravitational field strength on Earth is represented by a lowercase 𝑔. And an object with a mass 𝑚 in this gravitational field strength 𝑔 will have a weight force of 𝑊 equal to 𝑚 multiplied by 𝑔. This force will act vertically downward on the object toward the center of mass of Earth. Since the ball in this question is an object that will have some value of mass, then that ball must experience a downward force on it as a result of gravity at all points in time.
Notice that in diagram (C), there’s a red arrow drawn on the ball at every instant in time pointing vertically downward, and this red arrow represents the weight force due to gravity. However, this is clearly not the case in diagram (B). If we look at the horizontal part of the surface, we see no forces acting on the ball at all. And so that means that the weight force due to gravity is not being shown here. This diagram then can’t be correct, so we can eliminate answer option (B). This leaves us with the diagram shown in option (C). And let’s now clear our annotations of this diagram and take another look at it to check that it makes sense.
In diagram (C), we can see that at each instant in time, there are two forces shown acting on the ball. We’ve already identified that the red arrow, which points vertically downward at each point, represents the weight force due to gravity.
We’ve then also got the blue arrow, which we can see is at 90 degrees or perpendicular to the surface at each point. And this represents the normal reaction force of the surface on the ball. During the horizontal part of the ball’s motion when it’s moving at a constant speed, this normal reaction force points vertically upward and exactly balances the downward force due to gravity. During the other two sections when the ball is accelerating down the slope, this is not the case. During these sections of the motion, the blue arrow representing the reaction force is not in the opposite direction to the red arrow representing the weight force due to gravity.
That means that during these two parts of the motion where the ball is accelerating down the slope, the forces are not balanced and so there is a net force on the ball. This net force ends up acting in the direction down the slope, causing the ball to accelerate in this direction. Overall then, we can see that these forces shown in diagram (C) make complete sense in relation to the motion of the ball. Meanwhile, what’s shown in diagram (B) is actually the net force acting on the ball at each moment. Notice that this net force is zero on the horizontal surface and acts down the slope when the ball is accelerating.
Now, the question asks for the diagram showing all the correct forces acting on the ball at equal time intervals, not just the net force. And so we can identify our answer as diagram (C).
