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? [2A] [1C]
b) At the highest point of the swinging motion, what is the angle between the chains and their natural
resting position? [2T] [1C]
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, all of the child's energy is in the form of kinetic energy (KE), given by the equation KE = (1/2)mv^2, where m is the mass of the child (40 kg) and v is the velocity (5 m/s).
At the highest point of the swinging motion, the child's energy is all in the form of potential energy (PE), given by the equation PE = mgh, where h is the height above the ground.
Therefore, we can equate the initial kinetic energy to the final potential energy:
(1/2)mv^2 = mgh
Simplifying and solving for h, we get:
h = (1/2)(v^2)/g
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values, we have:
h = (1/2)(5^2)/(9.8)
h ≈ 1.275 m
Therefore, at the highest point of their swinging motion, the child is approximately 1.275 meters above the ground.
b) The angle between the chains and their natural resting position at the highest point of the swinging motion can be found using trigonometry. The chains form a right triangle with the vertical line from the attachment point to the child as the hypotenuse, and the vertical line from the attachment point to the highest point of the swing as one of the legs.
We can use the inverse trigonometric function to find the angle θ:
sin θ = opposite/hypotenuse
In this case, the opposite side is the length of the chain (2.1 m) and the hypotenuse is the height above the ground at the highest point (1.275 m).
Therefore:
sin θ = 2.1/1.275
θ = sin^(-1)(2.1/1.275)
θ ≈ 58.8 degrees
Therefore, at the highest point of the swinging motion, the angle between the chains and their natural resting position is approximately 58.8 degrees.