A 27.59 kg child is on a swing that hangs from 2.26-m-long chains. What is her maximum speed if she swings out a 48° angle?
4.15m/s
To find the maximum speed of the child, we can use the conservation of mechanical energy principle, which states that the total mechanical energy of a system remains constant if no external forces are acting on it.
The mechanical energy of a swinging child is the sum of its potential energy and kinetic energy.
1. Potential energy: When the child swings out at a given angle, her potential energy is at its maximum. We can calculate the potential energy using the formula: PE = mgh, where m is the mass (27.59 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height.
The height (h) can be found using trigonometry. The vertical distance between the lowest point and the highest point of the swing is given by h = L - Lcosθ, where L is the length of the chain (2.26 m) and θ is the angle of swing (48°).
2. Kinetic energy: When the child reaches the lowest point of the swing, her potential energy is zero and kinetic energy is at its maximum. We can calculate the kinetic energy using the formula: KE = (1/2)mv², where v is the velocity.
Since the total mechanical energy remains constant, we can equate the potential energy at the highest point (PE) to the kinetic energy at the lowest point (KE).
mgh = (1/2)mv²
Canceling out the mass:
gh = (1/2)v²
Rearranging the equation for velocity:
v = √(2gh)
Now, substitute the known values:
g = 9.8 m/s² (acceleration due to gravity)
h = L - Lcosθ = 2.26 m - 2.26 m * cos(48°)
Finally, calculate the maximum speed (v):
v = √(2 * 9.8 m/s² * 2.26 m * (1 - cos(48°)))
v ≈ 3.98 m/s
Therefore, the maximum speed of the child is approximately 3.98 m/s.