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 principle of conservation of mechanical energy.
a) At the bottom of the swinging motion, the child has only kinetic energy and no potential energy. At the highest point, the child will have only potential energy and no kinetic energy. Therefore, the total mechanical energy of the system (child + swing) is conserved.
The potential energy at the bottom is given by: PE_bottom = m*g*h_bottom
where m = mass of the child = 40 kg
g = acceleration due to gravity = 9.8 m/s^2
h_bottom = height above the ground at the bottom = 0.6 m
The kinetic energy at the bottom is given by: KE_bottom = (1/2)*m*v_bottom^2
where v_bottom = velocity at the bottom = 5 m/s
At the highest point, the potential energy is given by: PE_highest = m*g*h_highest
where h_highest = height above the ground at the highest point (to be determined)
Equating the total initial mechanical energy (KE_bottom + PE_bottom) to the total final mechanical energy (PE_highest), we have:
(1/2)*m*v_bottom^2 + m*g*h_bottom = m*g*h_highest
Rearranging the equation, we get:
h_highest = (1/2)*v_bottom^2/g + h_bottom
Substituting the given values into the equation:
h_highest = (1/2)*(5 m/s)^2/9.8 m/s^2 + 0.6 m
h_highest ≈ 1.28 m
Therefore, the child is approximately 1.28 m above the ground at the highest point of their swinging motion.
b) At the highest point, the chains are at their maximum angle with respect to their natural resting position, which is perpendicular to the ground. This means that the chains will form a right angle with the ground.
So, the angle between the chains and their natural resting position is 90 degrees.