A horizontal plank of uniform density is held in position by a downward-directed force at one of its ends as shown in the diagram. The mass of the plank is 25 kilograms. What is the size of the downward force?
Okay, so in this question, we can see that we’ve got a horizontal plank. That’s shown in the diagram here. We’re told that this plank has a uniform density. And it’s held in position by a downward-directed force at one of its ends. And we can already see the force in the diagram. It’s this force here. So first things first, let’s actually call it something. Let’s call this force 𝐹.
We’ll come back to this later because this is what we’ll be trying to find out in the question. Anyway moving on, we’re told that the mass of the plank is 25 kilograms. What we need to do as we said earlier is to work out the size of the downward force.
In other words, we need to work out what the value of 𝐹 is. So 𝐹 is equal to a question mark. We’re trying to work out what 𝐹 is. And we can also write down the other piece of information we’ve been given, which is that the mass of the plank is 25 kilograms.
Now why is the mass of the plank relevant to us? Well let’s look at the diagram. There’s a force 𝐹 acting on the left end of the plank. And the fulcrum, which is here, this triangle here is nowhere near the center of the plank. So how is the plank balancing? Why is it not turning anticlockwise because of the force 𝐹? Well, it’s because there has to be another force acting on the plank which hasn’t been shown in the diagram.
That’s the weight of the plank. We can recall that the weight of an object, 𝑊, is given by multiplying the mass of the object, 𝑚, by the gravitational field strength of the Earth, 𝑔. So we can work out what the weight of the plank is because we already know what the mass is. We’ve been told that the mass is 25 kilograms. And we know that the gravitational field strength of the Earth is 9.8 meters per second squared.
However, we won’t calculate the weight just yet. Instead, what we need to do is to work out where in the diagram the weight of the plank will be acting. Well, we know that the weight of an object acts from the center of its mass. So we need to work out where the center of mass of the plank is. Luckily in the question, we’ve been told that the plank has a uniform density. This means that the mass of the plank is evenly distributed throughout its entire volume.
In other words, if this was the plank, then it’s not like this half is heavier than this half here or vice versa or anything like that. The entire mass of the plank is evenly distributed along its length. The other thing is the plank is a nice shape for us to work with. In the diagram, it’s shown as rectangular. This means that the center of mass of the plank is simply going to be at the center of the plank. That’s here.
So we can now label the weight of the plank, because we know that the weight of the plank must act in a downward direction from the center of mass. And hence, this force is 𝑊. So now we can see that the force 𝐹 and the force 𝑊 are competing with each other in order to keep the plank stable. 𝑊 is trying to turn the plank clockwise, whereas 𝐹 is trying to turn the plank anticlockwise.
Now the turning forces applied by these two forces must be the same size but acting in opposite directions in order for the plank to be balanced like it is in the diagram. So to reiterate, we’re not saying that the force 𝐹 and the force 𝑊 are the same. Instead, what we’re saying is that the amount by which force 𝐹 wants to turn the plank anticlockwise is the same as the amount by which 𝑊 is turning the plank clockwise. It’s the turning forces that cancel each other out.
And these turning forces are known as torque. We calculate torque by multiplying the force applied by its perpendicular distance to the fulcrum. So let’s work out the turning forces caused by force 𝐹 and force 𝑊. Let’s call the first one 𝑇 sub anticlockwise for the anticlockwise torque. The anticlockwise torque is simply the force 𝐹 multiplied by the perpendicular distance between the force 𝐹 and the fulcrum, which is this distance here.
And let’s call this distance 𝑎 for now. So the anticlockwise torque is 𝐹𝑎. We can do the same thing for the clockwise torque. Calling this 𝑇 sub clockwise, we get that it’s equal to the force 𝑊 multiplied by the perpendicular distance to the fulcrum, which is this distance here. Let’s call this distance 𝑏. And so the clockwise torque is 𝑊 times 𝑏. Now, we’ve said that the sizes of these forces must be equal in order for the plank to be in equilibrium.
Of course the two torques act in opposite directions. One is trying to turn the plank clockwise and the other is trying to turn it anticlockwise. But the sizes of the torques must be the same. Therefore, what we have is that 𝐹𝑎 is equal to 𝑊𝑏. Now what we’re trying to do is to work out what the value of 𝐹 is. So let’s rearrange the equation. Dividing both sides of the equation by 𝑎, meaning that the 𝑎s on the left-hand side cancel out, we’re left with 𝐹 is equal to 𝑊𝑏 over 𝑎.
So, as we saw earlier, we can work out what 𝑊 is. And from the diagram, we already know what distance 𝑎 is. So let’s work out what distance 𝑏 is. But we said earlier that the weight acts from the center of mass of the plank. And the center of mass of the plank is simply the center of the plank. So from one end, let’s say from the left end, we know that the distance to the center of mass must be half the length of the plank.
So let’s call this distance 𝑐. And then we can say that distance 𝑐 is equal to the length of the plank, which is 90 centimeters, divided by two, because it’s half the length of the plank. This ends up being 45 centimeters. But we don’t need to know what 𝑐 is. We actually want to know what 𝑏 is. However, we can see that 𝑐 is equal to 𝑎 plus 𝑏.
And so 𝑏 is equal to 𝑐 minus 𝑎, which ends up being 45 centimeters minus 15 centimeters. And that simplifies to 30 centimeters. So now we found the value of 𝑏 and we already know the value of 𝑎, which means we can come back to our equation over here: 𝐹 is equal to 𝑊𝑏 over 𝑎. So we can say 𝐹 is equal to 𝑊, which we said earlier was 𝑚𝑔, multiplied by 𝑏 divided by 𝑎. And now let’s plug in all our values.
What we end up getting is 𝐹 is equal to 25, which is the mass, multiplied by 9.8, which is 𝑔, multiplied by 30 centimeters divided by 15 centimeters. Now normally, we’d want to convert the lengths 𝑎 and 𝑏 into meters because that’s a standard unit. But because we’re dividing one by the other, what we actually have is 30 centimeters divided by 15 centimeters.
And the units of centimeters cancel out. So we could have converted them both to meters, but it doesn’t really matter here. Anyway, let’s evaluate the fraction then. Plugging all the values into our calculator, we find that 𝐹 is equal to 490 newtons. And we can realize that if we had converted the values of 𝑎 and 𝑏 into meters, we still would have got the same answer. So now we have the final answer to our question: the size of the downward force 𝐹 is 490 newtons.