Jill of the Jungle swings on a vine 7.0 m long. What is the tension in the vine if Jill, whose mass is 65 kg, is moving at 2.3 m/s when the vine is vertical?

To find the tension in the vine, we can use the principle of conservation of energy. The initial gravitational potential energy of Jill at the top of her swing is equal to the final kinetic energy of Jill at the bottom of her swing.

The gravitational potential energy (PE) at the top of the swing is given by:
PE = m * g * h

where m is the mass of Jill, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height above the ground.

Since the vine is vertical at the bottom of the swing, the total mechanical energy remains constant. Therefore, the kinetic energy (KE) at the bottom of the swing is given by:
KE = (1/2) * m * v^2

where v is the velocity of Jill at the bottom of the swing.

Now, let's calculate the height Jill reaches at the top of the swing:
PE = m * g * h
65 * 9.8 * h = 65 * 9.8 * 7.0
h = 7.0 m

The height Jill reaches is equal to the length of the vine, 7.0 m.

Now, let's calculate Jill's kinetic energy at the bottom of the swing:
KE = (1/2) * m * v^2
KE = (1/2) * 65 * (2.3^2)
KE = 181.145 J

Since the total mechanical energy remains constant, the potential energy at the top is equal to the kinetic energy at the bottom:
PE = KE
m * g * h = (1/2) * m * v^2

Now, we can solve for the tension in the vine. At the bottom of the swing, Jill's weight is balanced by the tension in the vine:
Tension = Weight = m * g

Tension = 65 * 9.8
Tension = 637 N

Therefore, the tension in the vine is 637 Newtons.

When the vine is vertical, the tension equals M g + M V^2/R