Video Transcript
A body of mass four kilograms is hanging from a spring balance fixed to the ceiling of a lift. Given that the reading on the balance is 1042 gram-weight, determine the magnitude of the acceleration of the lift to the nearest two decimal places. Take 𝑔 to be equal to 9.8 meters per square second.
Let’s begin by modeling the free-body diagram here. We have this body of mass four kilograms hanging from a spring balance. Since it has a mass of four kilograms, the downwards force of its weight is four times acceleration due to gravity; it’s four 𝑔. We’re told that the reading on the balance is 1042 gram-weight. This is equivalent to thinking about the tension in the spring.
Now it’s worth noting that that four 𝑔, that weight force, is in newtons. This means we could do with converting the upward force from gram-weight into newtons. First, we know we can convert from gram-weight into kilogram-weight by dividing by 1000, so 1042 gram-weight is equivalent to 1.042 kilogram-weight. We also know that one kilogram-weight is equivalent to 9.8 newtons. So, we multiply 1.042 by 9.8, and we find the upward force is 10.2116 newtons.
With this in mind, we can then calculate the net force on the object. And the reason we want to do this is because we’re trying to determine the magnitude of the acceleration of the lift. So, we can use the formula net force is equal to mass times acceleration. We might make an assumption that the lift is accelerating downward. Now we make this assumption because four 𝑔, four times 9.8, is going to be greater than 10.2116. But of course, it doesn’t matter if we make an assumption that the lift is accelerating in the opposite direction. We’ll just determine the final acceleration by looking at the sign.
Since the net force is equal to mass times acceleration, we can say that four 𝑔 minus 10.2116 is equal to four 𝑎. In other words, four times 9.8 minus 10.2116 is equal to four 𝑎. Calculating the left-hand side and we get 28.9884. And then we can solve to find the acceleration in the system by dividing through by four. So, acceleration is 28.9884 divided by four. That gives us 7.2471 meters per square second. Correct to two decimal places, that’s 7.25. So, the magnitude of the acceleration of the lift is 7.25 meters per square second.
