2. 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 conservation of mechanical energy. At the bottom of the swinging motion, all of the child's gravitational potential energy is converted into kinetic energy. At the highest point of the swinging motion, all of the child's kinetic energy is converted back into gravitational potential energy.
The gravitational potential energy of the child is given by the equation:
PE = mgh
Where m is the mass of the child (40 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height above the ground.
At the bottom of the swing, the child is 0.6 m above the ground and traveling at a speed of 5 m/s. Therefore, we can calculate the kinetic energy at the bottom of the swing using the equation:
KE = 0.5mv^2
Where v is the velocity of the child.
Since energy is conserved, the kinetic energy at the bottom of the swing is equal to the gravitational potential energy at the highest point of the swing:
0.5mv^2 = mgh
Simplifying the equation:
0.5(40 kg)(5 m/s)^2 = 40 kg * 9.8 m/s^2 * h
100 J = 392 h
h = 100 J / 392 J/m
h ≈ 0.255 m
Therefore, the child is approximately 0.255 m above the ground at the highest point of their swinging motion.
b) To find the angle between the chains and their natural resting position at the highest point of the swinging motion, we can use trigonometry. The chains form a right-angled triangle with the vertical line passing through the highest point of the swing as the hypotenuse, the chains as one side, and the horizontal line connecting the highest point of the swing to the pivot as the other side.
Using the lengths of the chains (2.1 m) as the side opposite the angle and the hypotenuse, we can use the sine function to find the angle:
sin(angle) = (chain length opposite angle) / (hypotenuse)
sin(angle) = 2.1 m / 2.1 m
sin(angle) = 1
Since the sine of an angle can only be between -1 and 1, this means the angle is 90 degrees.
Therefore, at the highest point of the swinging motion, the angle between the chains and their natural resting position is 90 degrees.