A 29 kg child sits in a swing supported by
two chains, each 4.2 m long.
If the tension in each chain at the lowest
point is 213 N, find the child’s speed at the
lowest point. (Neglect the mass of the seat.)
To find the child's speed at the lowest point in the swing, we can use the principle of conservation of energy. At the highest point, the child's potential energy is at its maximum, and at the lowest point, the potential energy is converted to kinetic energy.
First, let's find the potential energy at the highest point. The potential energy (PE) can be calculated using the equation:
PE = m * g * h
where m is the mass of the child (29 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height from the highest point to the lowest point. In this case, since we are neglecting the mass of the seat, h is equal to the length of each chain (4.2 m).
PE = 29 kg * 9.8 m/s² * 4.2 m
= 1200.36 J
Next, at the lowest point, all the potential energy is converted to kinetic energy. The kinetic energy (KE) formula is given by:
KE = (1/2) * m * v²
where m is the mass of the child (29 kg), and v is the velocity (speed) of the child at the lowest point.
Since the potential energy at the highest point is converted entirely into kinetic energy, we can equate the two:
PE = KE
This gives us:
29 kg * 9.8 m/s² * 4.2 m = (1/2) * 29 kg * v²
Simplifying further:
1200.36 J = 0.5 * 29 kg * v²
Now we can solve for v:
v² = (2 * 1200.36 J) / 29 kg
v² = 82.7949 m²/s²
Taking the square root of both sides gives us the velocity:
v = √(82.7949 m²/s²)
v ≈ 9.10 m/s
Therefore, the child's speed at the lowest point is approximately 9.10 m/s.