A 29 kg child sits in a swing supported by two
chains, each 4.2 m long.
If the tension in each chain at the lowest
point is 213 N, find the child’s speed at the
lowest point. (Neglect the mass of the seat.)
Answer in units of m/s.
child weight (w) = 29 g
centripetal force = m v^2 / r = (2 * 213) - w
solve for v
To find the child's speed at the lowest point, we can use the principle of conservation of energy. At the highest point, the child's potential energy is maximum and all converted to kinetic energy at the lowest point.
Let's break down the steps to solve the problem:
Step 1: Calculate the potential energy at the highest point:
The potential energy at the highest point is equal to the total mass of the child multiplied by the acceleration due to gravity (9.8 m/s^2) multiplied by the height of the swing.
Potential energy (PE) = mass (m) * gravity (g) * height (h)
PE = 29 kg * 9.8 m/s^2 * 4.2 m
Step 2: Calculate the kinetic energy at the lowest point:
Using the principle of conservation of energy, we know that the potential energy at the highest point is equal to the kinetic energy at the lowest point.
PE = KE
So, PE = m * g * h = KE
Step 3: Calculate the speed at the lowest point:
The kinetic energy (KE) at the lowest point is equal to one-half of the mass (m) multiplied by the square of the velocity (v).
KE = (1/2) * m * v^2
Set the potential energy equal to the kinetic energy:
PE = KE
m * g * h = (1/2) * m * v^2
Simplify and solve for velocity (v):
2gh = v^2
v = sqrt(2gh)
Step 4: Plug in the values and calculate the child's speed:
v = sqrt(2 * 9.8 m/s^2 * 4.2 m)
v ≈ sqrt(82.32) ≈ 9.08 m/s
Therefore, the child's speed at the lowest point is approximately 9.08 m/s.