A trapdoor on a stage has a mass of 20.8 kg and a width of 1.51 m (hinge side to handle side). The door can be treated as having uniform thickness and density. A small handle on the door is 1.41 m away from the hinge side. A rope is tied to the handle and used to raise the door. At one instant, the rope is horizontal, and the trapdoor has been partly opened so that the handle is 1.13 m above the floor. What is the tension, T, in the rope at this time?

Question

A trapdoor on a stage has a mass of 20.8 kg and a width of 1.51 m (hinge side to handle side). The door can be treated as having uniform thickness and density. A small handle on the door is 1.41 m away from the hinge side. A rope is tied to the handle and used to raise the door. At one instant, the rope is horizontal, and the trapdoor has been partly opened so that the handle is 1.13 m above the floor. What is the tension, T, in the rope at this time?

Answer 1

To find the tension, T, in the rope at this time, we can use the principle of moments. Moments can be calculated by multiplying force by distance. In this case, the force is the tension in the rope, and the distance is the distance from the hinge to the handle.

First, let's understand the setup. The trapdoor is partially opened, and the handle is 1.13 m above the floor. The handle is 1.41 m away from the hinge side. To find the tension in the rope, we can set up an equation based on the principle of moments.

The principle of moments states that the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about the same point, assuming the system is in equilibrium.

In this case, we can choose the hinge point as our reference point. So the clockwise moments will be positive, and the anticlockwise moments will be negative.

Let's denote the tension in the rope as T and the width of the trapdoor as w. We know the mass (m) of the trapdoor is 20.8 kg, and the width (w) is given as 1.51 m.

The moment due to the weight of the trapdoor can be calculated as (m * g), where g is the acceleration due to gravity (approximately 9.8 m/s^2). The force will act at the center of mass of the trapdoor, which is halfway between the hinge side and the handle side. So the distance from the hinge side to the center of mass is (w / 2).

The moment due to the tension in the rope will be (T * distance from hinge to handle side). In this case, the distance is (1.41 m - 1.13 m) = 0.28 m.

Now, let's set up the equation using the principle of moments:

(m * g) * (w / 2) = T * (1.41 m - 1.13 m)

Substituting the given values:

(20.8 kg * 9.8 m/s^2) * (1.51 m / 2) = T * 0.28 m

This gives us:

(202.24 N) * (0.755 m) = T * 0.28 m

Simplifying:

T * 0.28 m = 152.48 N

T = 152.48 N / 0.28 m

T ≈ 545.43 N

So, the tension in the rope at this time is approximately 545.43 N.