A 45.7-kg boy on a swing moves in a circular arc of radius 3.80 m.At the lowest position, the child's speed reaches 2.78 m/s. Determine the magnitude of the tension in each of the two vertical support chains.
i used mv^2/r, but i keep getting the answer wrong. the answer is suppose to be 270 m/s
To determine the magnitude of the tension in each of the two vertical support chains, you can use the following steps:
Step 1: Calculate the centripetal force acting on the boy.
The centripetal force is necessary to keep the boy moving in a circular arc. The formula for centripetal force is given by F = m * v^2 / r, where F is the centripetal force, m is the mass of the boy, v is the velocity, and r is the radius.
Given that the mass of the boy is 45.7 kg, the velocity is 2.78 m/s, and the radius is 3.80 m, you can substitute these values into the equation:
F = (45.7 kg) * (2.78 m/s)^2 / (3.80 m)
F = (45.7 kg) * (7.7284 m^2/s^2) / (3.80 m)
F = 93.9316 kg.m/s^2 / 3.80 m
F ≈ 24.71 kg.m/s^2
Step 2: Distribute the centripetal force evenly between the two support chains.
Since there are two support chains, the total force is equally distributed between them. Therefore, the tension force in each support chain is half of the centripetal force.
T = (24.71 kg.m/s^2) / 2
T ≈ 12.36 kg.m/s^2
Step 3: Convert the tension force from kg.m/s^2 to newtons (N).
Since the unit of force is newtons, you can convert the tension from kg.m/s^2 to newtons by dividing by the acceleration due to gravity (standard value of 9.8 m/s^2).
T = 12.36 kg.m/s^2 / 9.8 m/s^2
T ≈ 1.26 N
Therefore, the magnitude of tension in each of the two vertical support chains is approximately 1.26 N.
To find the magnitude of the tension in each of the two vertical support chains, you'll need to consider the forces acting on the boy when he is at the lowest position of the swing.
At the lowest position, the boy is experiencing two forces:
1. The force due to gravity, which is equal to his weight (mg) and acts vertically downwards.
2. The tension in each of the two vertical support chains, which act vertically upwards.
To find the tension, you can set up the equation of forces:
Tension - Weight = Centripetal Force
The centripetal force is given by the formula mv^2/r, where m is the mass of the boy, v is his speed, and r is the radius of the circular arc.
Let's calculate the centripetal force first:
Centripetal Force = mv^2/r
= (45.7 kg)(2.78 m/s)^2 / 3.8 m
≈ 92.085 kg·m/s^2
Now, plug this value back into the equation:
Tension - Weight = 92.085 kg·m/s^2
The weight can be calculated by multiplying the mass (45.7 kg) by the acceleration due to gravity (9.8 m/s^2):
Weight = mg
= (45.7 kg)(9.8 m/s^2)
≈ 448.86 kg·m/s^2
Now, rearrange the equation to solve for the tension:
Tension = 92.085 kg·m/s^2 + 448.86 kg·m/s^2
= 540.945 kg·m/s^2
Since there are two vertical support chains, divide this value by 2:
Tension in each chain = 540.945 kg·m/s^2 / 2
≈ 270 kg·m/s^2
Therefore, the magnitude of the tension in each of the two vertical support chains is approximately 270 kg·m/s^2.