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?
a) To find the height of the child at the highest point of their swinging motion, we can use the principle of conservation of mechanical energy. At the bottom of the swinging motion, the child has only kinetic energy (due to their speed) and no potential energy (as they are at the lowest point). At the highest point, the child has only potential energy and no kinetic energy.
We can equate the initial kinetic energy to the final potential energy:
(1/2)mv² = mgh
where m is the mass of the child (40 kg), v is the speed of the child at the bottom (5 m/s), g is the acceleration due to gravity (9.8 m/s²), and h is the height of the child at the highest point.
Solving for h, we have:
h = (1/2)v²/g
= (1/2)(5²)/(9.8)
≈ 1.275 m
Therefore, the child is approximately 1.275 meters above the ground at the highest point of their swinging motion.
b) At the highest point of the swinging motion, the angle between the chains and their natural resting position is equal to the angle of the swing's arc. This angle can be determined using trigonometry.
We can consider the swing as a pendulum, where the chains act as the length of the pendulum. The angle θ is related to the length of the pendulum and the vertical displacement of the child by the equation:
sin(θ) = h/L
where L is the length of the chains (2.1 meters). Rearranging the equation, we have:
θ = arcsin(h/L)
= arcsin(1.275/2.1)
≈ 34.49°
Therefore, at the highest point of the swinging motion, the angle between the chains and their natural resting position is approximately 34.49 degrees.