A 31.8 kg child swings on a rope with a length of 6.12 m that is hanging from a tree. At the bottom of the swing, the child is moving at a speed of 4.2 m/s. What is the tension in the rope?
Add the child's weight (M g) to M V^2/R, the centripetal force required by the motion.
To find the tension in the rope, we can use the principle of conservation of energy. The total mechanical energy of the system remains constant throughout the swing.
First, let's calculate the potential energy (PE) and kinetic energy (KE) of the child at the bottom of the swing.
The potential energy at the bottom of the swing can be calculated using the formula:
PE = m * g * h
Where m is the mass of the child (31.8 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height of the swing.
Since the child is at the bottom of the swing, h = 0. Therefore, the potential energy is zero.
Next, let's calculate the kinetic energy at the bottom of the swing using the formula:
KE = (1/2) * m * v^2
Where m is the mass of the child (31.8 kg) and v is the velocity of the child at the bottom of the swing (4.2 m/s).
Plugging in the values, we get:
KE = (1/2) * 31.8 kg * (4.2 m/s)^2 = 353.808 J
Since the total mechanical energy is conserved, the kinetic energy at the bottom of the swing is equal to the potential energy at the top of the swing.
Now, let's calculate the tension in the rope at the bottom of the swing. The tension in the rope provides the centripetal force keeping the child in circular motion.
The centripetal force can be calculated using the formula:
F = m * a
Where m is the mass of the child (31.8 kg) and a is the centripetal acceleration.
The centripetal acceleration can be calculated using the formula:
a = v^2 / r
Where v is the velocity (4.2 m/s) and r is the radius of the swing (the length of the rope, 6.12 m).
Plugging in the values, we get:
a = (4.2 m/s)^2 / 6.12 m = 2.883 m/s²
Now, let's calculate the tension using the formula:
F = m * a
Plugging in the values, we get:
F = 31.8 kg * 2.883 m/s² = 91.6734 N
Therefore, the tension in the rope is approximately 91.67 N.