A person is to be released from rest on a swing pulled away from the vertical by an angle of 15.8°. The two frayed ropes of the swing are 2.75 m long, and will break if the tension in either of them exceeds 350 N.
What is the maximum weight the person can have and not break the ropes?
I started by using the equation 350N/cos15.8 = Weight in one rope (2) = 727.5. This answer was incorrect. I am not sure if I have the picture drawn wrong. Where exactly does the length of the ropes come in? Please help!
Max tension will occur at the bottome, and tension will equal weight plus centripetal force. Veloicty at the bottom is dependent on the height, when the swing started: here height was 2.75(1-cosTheta).
Tension= weight+ mv^2/r
and mv^2=2mgh, with h above.
To determine the maximum weight the person can have without breaking the ropes, we need to consider the tension in the ropes at the bottom of the swing. At the bottom, the tension in the ropes will be at its maximum.
Let's break down the problem step by step:
1. Start by drawing a diagram to visualize the situation. Draw a swing with the person sitting in it, and the ropes attached at the top point. The ropes make an angle of 15.8° with the vertical.
2. We know that the length of each rope is 2.75 m. This information is crucial because we need to find the velocity at the bottom of the swing, which depends on the height from which the swing started.
3. The height (h) from which the swing started can be calculated as follows:
h = length of rope × (1 - cosθ)
h = 2.75 m × (1 - cos(15.8°))
4. Now that we have the height, we can calculate the velocity (v) at the bottom of the swing using the conservation of energy:
mgh = (1/2)mv²
v = √(2gh)
v = √(2 × g × h)
Note: In this equation, m represents the mass of the person, g represents the acceleration due to gravity (9.8 m/s²), and h is the height calculated in the previous step.
5. Now we can calculate the tension (T) at the bottom of the swing using the centripetal force formula:
T = weight + (m × v²) / r
Here, weight represents the mass (m) of the person multiplied by the acceleration due to gravity (9.8 m/s²), and r is the radius of rotation, which is equal to the length of the rope (2.75 m).
6. Since the ropes will break if the tension exceeds 350 N, we can set up the following equation to find the maximum weight (m):
T ≤ 350 N
7. Substitute the values into the equation and solve for mass:
350 N ≥ (m × g) + (m × v²) / r
Note: You mistakenly used cosine instead of calculating the actual height and velocity, which led to an incorrect answer.
8. Once you solve the equation, you will find the maximum weight (mass) that the person can have without breaking the ropes.
By following these steps, you should be able to determine the correct maximum weight that the person can have.