Question Video: Mass, Weight, and Applied Force

A climber is standing at the base of a cliff, holding onto a rope that is attached to the top of the cliff. The climber is pulling on the rope in order to pull himself off from the ground, but his feet stay in contact with the ground, just not supporting the climber’s full weight. The climber’s mass is 50 kg and the rope exerts an upward vertical force of 150 N on the climber. What is the force exerted on the ground beneath the climber due to his weight?

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Video Transcript

A climber is standing at the base of a cliff holding onto a rope that is attached to the top of the cliff. The climber is pulling on the rope in order to pull himself off from the ground, but his feet stay in contact with the ground, just not supporting climber’s full weight. The climber’s mass is 50 kilograms and the rope exerts an upward vertical force of 150 newtons on the climber. What is the force exerted on the ground beneath the climber due to his weight?

Okay, so in this situation first of all, we’ve got a cliff. And then we’ve got a rope which is attached to the top of the cliff and is being let down. We’ve also a got a mountain climber who is holding onto the rope and actually pulling on it. Now, as this mountain climber pulls down on the rope, we can use Newton’s third law of motion to see that the rope exerts an equal and opposite upward force on the climber. Newton’s third law of motion, by the way, says that if an object, let’s say object A, exerts a force on object B, that’s another object, then the second object, object B, will exert an equal and opposite force on object A.

And that’s exactly what we have going on here. The mountain climber is pulling down on the rope. And the rope is pulling an equal and opposite force on the mountain climber. In other words, the climber is being pulled upwards. Now, we’ve been told that this upward force on the climber due to the rope is enough to slightly raise the climber off the ground. But his feet still stay in contact with the ground. However, they aren’t supporting the climbers’ full weight. As well as this, we’ve been told that the rope exerts an upward vertical force of 150 newtons. And so we can label this force, 150 newtons, as the upward force exerted by the rope on the climber.

Another thing we’ve been told in the question is that the climbers mass is 50 kilograms. And so based on this information, we can work out the weight of the climber. But before we do that, let’s read the final sentence of the question. It says, what is the force exerted on the ground beneath the climber due to his weight? In other words, we’re not actually asked to find the weight of the climber. We’ve been asked to find the force exerted on the ground beneath the climber due to his weight.

So if we look more closely at our climber, let’s label the 150-newton force exerted by the rope on the climber, as well as the weight of the climber which will always act in a downward direction. Now, obviously both of these forces would ideally be placed on the body of the climber. But just for clarity, we’ve drawn it to the side. But anyway, so the weight of the climber is something that we don’t know yet. So let’s just call it 𝑤.

Now once again, we’re not being asked to find the weight of the climber. We will have to during the course of this question. But that’s not what the question is asking for. But the fact of the matter is that there are two forces acting on the climber, the 150-newton upward force and the downward weight force. We have enough information in the question to calculate what the weight force is because we’ve been given the mass of the climber. So let’s do that. We can recall that the weight of any object is given by multiplying the mass of that object by the acceleration due to gravity on Earth or, in other words, the gravitational field strength on Earth.

We can also recall that, on earth, the gravitational field strength is 9.8 metres per second squared. So let’s calculate the weight of the climber. We said that the weight of the climber, 𝑤, is equal to the mass of the climber, 50 kilograms, multiplied by the gravitational field strength, which is 9.8 metres per second squared. And because we’re working in standard units, kilograms for mass and metres per second squared for gravitational field strength, our answer for the weight is going to be in its own standard unit as well. And weight is a force, so the standard unit is newtons.

So we can evaluate the right-hand side to find that the weight of the climber is 490 newtons. And hence, we can replace the 𝑤 down here with 490 newtons. Now at this one, we’ve just figured out the values of both of the forces acting on the climber. The downward force is 490 newtons. And the upward force is 150 newtons. Well in this case, because the two forces are acting in opposite directions, they are going to oppose each other. And because the downward force is larger, that force is going to win.

So the overall effect on the climber, the resultant force as we call it, is going to be in a downward direction and is going to be 490 newtons minus 150 newtons because that is what will be the overall effect on the climber when we account for both of the forces acting on the climber. And that overall result is going to be 340 newtons and, as we said, is going to be in the downward direction because the downward force is larger by 340 newtons.

So let’s call this resultant overall force 𝐹. And at this point, we can actually replace the two forces acting on the climber with the resultant or overall force. And the reason that we can do this is because having two forces acting in the opposite directions, where one of them is downwards at 490 newtons and the other is upwards at 150 newtons, is exactly equivalent, is exactly the same as having just one downward force of 340 newtons acting on the climber.

So let’s pause and think for a minute. What have we just done. Well, what we’ve done is calculated the resultant or overall force acting on the climber. But why is that any use to us. Well it’s because if there is a downward force acting on the climber and the climber’s feet are touching the ground, then the climber is basically going to be pushing downwards onto the ground with a force of 340 newtons. And as we saw already, that force of 340 newtons was because of the two forces acting on the climber, the weight of the climber and also the force exerted by the rope on the climber.

Hence, this 340 newtons is the force exerted on the ground due to the weight of the climber, as well as due to the weight of the rope on the climber. So this is where we need to be careful. If the question is asking us for the force exerted on the ground due to the weight rather than the weight itself, then we need to think carefully about what that means. In this case, that meant finding the resultant or overall force on the climber and, therefore, the force that the climber exerts on the ground because the climber’s feet are in contact with the ground. So our final answer is that the force exerted on the ground beneath the climber due to his weight is 340 newtons.

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