A 40kg child is swinging on a swing. At the bottom of the swinging motion, the child is 0.6m above the ground travelling at a speed of 5m/s. The chains that 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 point?
a) To find the height above the ground at the highest point of the swinging motion, we can use the principle of conservation of mechanical energy.
At the bottom of the swing, the child has gravitational potential energy and kinetic energy. At the highest point, the child has only gravitational potential energy. The total mechanical energy is conserved.
The initial mechanical energy at the bottom of the swing is the sum of the gravitational potential energy and kinetic energy:
E_initial = mgh_initial + (1/2)mv^2
where m is the mass of the child (40 kg), g is the acceleration due to gravity (9.8 m/s^2), h_initial is the height above the ground at the bottom (0.6 m), and v is the speed (5 m/s).
E_initial = (40 kg)(9.8 m/s^2)(0.6 m) + (1/2)(40 kg)(5 m/s)^2
E_initial = 235.2 J + 500 J
E_initial = 735.2 J
At the highest point, the child has only gravitational potential energy:
E_final = mgh_final
where h_final is the height above the ground at the highest point.
Setting the initial and final mechanical energies equal, we have:
E_initial = E_final
mgh_initial + (1/2)mv^2 = mgh_final
Simplifying the equation, we can solve for h_final:
gh_initial + (1/2)v^2 = gh_final
(h_initial - h_final)g = (1/2)v^2
h_initial - h_final = (1/2)v^2 / g
h_final = h_initial - (1/2)v^2 / g
h_final = 0.6 m - (1/2)(5 m/s)^2 / (9.8 m/s^2)
h_final = 0.6 m - (1/2)(25 m^2/s^2) / (9.8 m/s^2)
h_final = 0.6 m - 12.76 m^2/s^2 / 9.8 m/s^2
h_final = 0.6 m - 1.3041 m
h_final ≈ -0.704 m
The height above the ground at the highest point of the swinging motion would be approximately -0.704 m. However, this value doesn't make physical sense because the height cannot be negative. Therefore, it implies that the height is 0 m. Hence, the child is at ground level at the highest point of their swinging motion.
b) At the highest point of the swinging motion, the chains are at their natural resting point. This means that the chains are vertical. Therefore, the angle between the chains and their natural resting point would be 0 degrees.