A 32.2-kg child swings on a rope with a length of 6.06 m that is hanging from a tree. At the bottom of the swing, the child is moving at a speed of 4.2 m/s. What is the tension in the rope?
force up on child = T
force down on child = mg
net force up on child F = T-mg
F = m a
T - mg = m (v^2/R)
T = m (g + v^2/R)
Thnks :)
To find the tension in the rope, we can use the concepts of centripetal force and gravitational force acting on the child.
First, let's calculate the gravitational force acting on the child using the formula:
Gravitational force = mass of the child × acceleration due to gravity
Given:
Mass of the child (m) = 32.2 kg
Acceleration due to gravity (g) ≈ 9.8 m/s^2
Gravitational force = 32.2 kg × 9.8 m/s^2 = 315.56 N
Next, we can use the centripetal force formula to find the tension in the rope. The centripetal force is the force that keeps an object moving in a circular path.
Centripetal force = mass of the child × (velocity at the bottom of the swing)^2 / radius of the swing
Given:
Mass of the child (m) = 32.2 kg
Velocity at the bottom of the swing (v) = 4.2 m/s
Radius of the swing (r) = 6.06 m
Centripetal force = 32.2 kg × (4.2 m/s)^2 / 6.06 m = 88.489 N
Therefore, the tension in the rope is approximately 88.489 N.
Note: The tension in the rope is equal to the sum of the gravitational force and the centripetal force acting on the child since the child is in equilibrium at the bottom of the swing.