Question

Logan is standing on a dock holding onto a rope swing that is =4.10 m long and suspended from a tree branch above. The rope is taut and makes a 30.0∘ angle with the vertical. Logan swings in a circular arc, passing through the bottom of the arc and then releasing the rope when it makes an angle of =13.1∘ with the perpendicular.

If Logan's mass is 79.0 kg how much work grav does gravity do on him up to the point where he releases the rope?

Answer 1

2025 J

Step-by-step explanation:

The work done by gravity on Logan can be calculated using the formula W = mgh, where W is the work done by gravity, m is the mass of Logan, g is the acceleration due to gravity and h is the change in height.

First we need to calculate the change in height. The initial height of Logan can be calculated using trigonometry. The vertical component of the rope length when it makes a 30 degree angle with the vertical is 4.10m * cos(30) = 3.55m.

Similarly, when Logan releases the rope at an angle of 13.1 degrees with the perpendicular (or 90-13.1=76.9 degrees with the vertical), his height above water level will be 4.10m * cos(76.9) = 0.93m.

So his change in height will be 3.55m - 0.93m = 2.62m.

Now we can calculate how much work gravity does on him: W = mgh = (79kg)(9.8 m/s^2)(2.62m) ≈ 2025 J.

So gravity does about 2025 J of work on Logan up to the point where he releases the rope.