A 40kg child is swinging on a swing. At the bottom of the swinging motion, the child is 0.6m above the ground and travelling at a speed of 5m/s. The chains the swing is attached to are 2.1m long.
a) How high above the ground is the child at the highest point of their swinging motion?
b) At the highest point of the swinging motion, what is the angle between the chains and their natural resting position?
To solve this problem, we can use the principles of conservation of mechanical energy and circular motion.
a) At the bottom of the swinging motion, the child has maximum kinetic energy and minimum potential energy. At the highest point of the swinging motion, the child has maximum potential energy and minimum kinetic energy. Therefore, we can equate the kinetic and potential energies at these two points:
1/2 * mv_1^2 + mgh_1 = 1/2 * mv_2^2 + mgh_2
Where:
m = mass of the child = 40 kg
v_1 = velocity at the bottom = 5 m/s
h_1 = height at the bottom = 0.6 m
v_2 = velocity at the top (which is 0 m/s)
h_2 = height at the top (unknown)
Substituting the given values and solving for h_2:
1/2 * 40 kg * (5 m/s)^2 + 40 kg * 9.8 m/s^2 * 0.6 m = 1/2 * 40 kg * (0 m/s)^2 + 40 kg * 9.8 m/s^2 * h_2
400 J + 235.2 J = 0 J + 384 J * h_2
635.2 J = 384 J * h_2
h_2 = 635.2 J / (384 J)
h_2 ≈ 1.654 m
Therefore, the child is approximately 1.654 meters above the ground at the highest point of their swinging motion.
b) The angle between the chains and their natural resting position can be determined by considering the geometry of the swing motion. When the child is at the highest point, the motion is in equilibrium and the gravitational force acting on the child is balanced by the tension in the swing chains.
Using Newton's second law in the vertical direction:
-T * cosθ + mg = 0
Where:
T = tension in the swing chains
θ = angle
Since the swing is in equilibrium, we can set T equal to the weight of the child:
T = mg
Substituting this into the equation above and solving for θ:
-mg * cosθ + mg = 0
mg * (1 - cosθ) = 0
cosθ = 1
θ = arccos(1)
θ = 0 radians
Therefore, the angle between the chains and their natural resting position at the highest point is 0 radians, which means the chains are vertical.