Video: Finding the Current Weight of a Body Suspended from the Ceiling of an Accelerating Lift by a Spring

A body of mass 32 kg was suspended from a spring balance fixed to the ceiling of an elevator. Given that the elevator was accelerating upward at 405 cm/s², find the apparent weight of the body. Take 𝑔 = 9.8 m/s².

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Video Transcript

A body of mass 32 kilograms was suspended from a spring balance fixed to the ceiling of an elevator. Given that the elevator was accelerating upward at 405 centimeters per square second, find the apparent weight of the body. Take 𝑔 to be equal to 9.8 meters per square second.

Let’s begin by sketching a diagram. We have a body of mass 32 kilograms suspended from a spring balance. That spring balance is fixed to the ceiling of an elevator. And so there are a couple of forces at play here. Firstly, there’s the downwards force of the body. That force is the weight, and it’s equal to mass times gravity, so 32 times 𝑔. Now, since the body itself is exerting a downward force on the spring balance, Newton’s third law tells us that the spring balance must also exert an upward force on the body itself. Let’s call that force tension, and actually that tension will give us the reading and the apparent weight of the body.

We know that the elevator itself is accelerating upward at 405 centimeters per square second. But actually, since there are 100 centimeters in a meter, we’re going to divide this value by 100 to get 4.05 meters per square second. And that’s because we’re currently working in kilograms. We’ve got gravity in meters per second, and so our final answer is going to be in newtons. And so we need to be consistent with our units throughout. Now, actually, the spring balance itself will exert a force on the ceiling of the elevator. But we’re not interested in that system of forces. And so we move on, and we use Newton’s second law of motion, force is equal to mass times acceleration.

We’re going to work out the net sum of the forces in our system, and then we’re going to set that equal to the mass of the body times the acceleration. And so we need to choose a positive direction. Let’s take upwards to be positive, since that’s the direction in which the body is moving. And since we have 𝑇 tension acting upward and 32𝑔 acting in the negative direction, the net sum on our system is 𝑇 minus 32𝑔. That’s equal to the mass of the body times acceleration. So that’s 32 times 4.05. We’re trying to find 𝑇 since that will tell us the apparent weight of the body. And so we’re going to add 32𝑔 to both sides. 𝑇 is 32 times 4.05 plus 32𝑔, which is equal to 32 times 4.05 plus 32 times 9.8. That gives us a value of 443.2. The apparent weight of the body then is 443.2 newtons.

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