To solve this problem, we can use the conservation of mechanical energy.
(a) To find the speed when the string is at 60° to the vertical, we need to determine the height at that point. We know that the stone's speed is maximum at the lowest point, so the mechanical energy is conserved.
The total mechanical energy at the lowest point is given by the sum of the kinetic energy and potential energy:
E = KE + PE = 1/2 * m * v^2 + m * g * h
Where:
m = mass of the stone = 2.1 kg
v = speed of the stone at the lowest point = 7.9 m/s
g = acceleration due to gravity = 9.8 m/s^2
h = height at the lowest point (which is unknown at this stage)
At the lowest point, the potential energy is zero (as given in the problem statement), so we can write:
E = 1/2 * m * v^2
Now, at the position when the string makes an angle of 60° to the vertical, the height is given by:
h = length of the string * (1 - cos(60°))
Substituting the known values, we have:
h = 3.6 m * (1 - cos(60°))
Solving this equation, we find:
h ≈ 3.6 m * (1 - 0.5) ≈ 1.8 m
Now, we can find the speed when the string is at 60° to the vertical using the conservation of mechanical energy:
E = 1/2 * m * v^2
v = sqrt(2 * E / m)
Substituting the known values, we have:
v = sqrt(2 * 1/2 * m * v^2 / m)
v = sqrt(v^2)
v ≈ v
Therefore, the speed when the string is at 60° to the vertical is approximately equal to the speed at the lowest point, which is 7.9 m/s.
(b) To find the greatest angle with the vertical that the string will reach during the stone's motion, we can use the conservation of mechanical energy again.
At the highest point reached by the stone, the speed is zero, so the mechanical energy is conserved between the lowest and highest points.
E = KE + PE = 1/2 * m * v^2 + m * g * h
At the highest point, the kinetic energy is zero, so we have:
E = PE = m * g * h
The potential energy at the highest point is equal to the potential energy at the lowest point (since the potential energy is constant throughout the motion).
m * g * h at the lowest point ≈ m * g * h at the highest point
m * g * (length of the string - h) ≈ m * g * h
(length of the string - h) ≈ h
2h ≈ length of the string
h ≈ (1/2) * length of the string
h ≈ (1/2) * 3.6 m ≈ 1.8 m
So, the greatest angle with the vertical that the string will reach during the stone's motion is approximately 90° - 60° = 30°.
(c) The total mechanical energy of the system can be found by considering the conservation of mechanical energy at any point in the motion.
From part (a), we found that the mechanical energy at the lowest point is given by:
E = 1/2 * m * v^2
Since the potential energy at the highest point is equal to the potential energy at the lowest point, we can calculate the total mechanical energy as:
Total mechanical energy = E + m * g * h
Substituting the known values, we have:
Total mechanical energy = 1/2 * m * v^2 + m * g * h
Total mechanical energy = 1/2 * 2.1 kg * (7.9 m/s)^2 + 2.1 kg * 9.8 m/s^2 * 1.8 m
Evaluating this expression, we find:
Total mechanical energy ≈ 88.762 J