A pendulum consists of a 2.6 kg stone swinging on a 4.5 m string of negligible mass. The stone has a speed of 7.9 m/s when it passes its lowest point. (a) What is the speed when the string is at 53 ¢ª to the vertical? (b) What is the greatest angle with the vertical that the string will reach during the stone's motion? (c) If the potential energy of the pendulum-Earth system is taken to be zero at the stone's lowest point, what is the total mechanical energy of the system?
To solve this problem, we need to use the principles of conservation of energy and the laws of motion for a pendulum.
(a) To find the speed when the string is at an angle of 53 ¢ª to the vertical, we can start by finding the potential energy at the lowest point and at the given angle.
The potential energy at the lowest point (when the stone is at its highest speed) can be calculated using the formula for gravitational potential energy:
Potential Energy = mass × acceleration due to gravity × height
Potential Energy at the lowest point = 2.6 kg × 9.8 m/s^2 × 4.5 m
Now, to find the velocity at an angle of 53 ¢ª to the vertical, we can equate the potential energy at both points with the kinetic energy at the given angle.
Potential Energy at the lowest point = Kinetic Energy at 53 ¢ª angle
2.6 kg × 9.8 m/s^2 × 4.5 m = (1/2) × mass × velocity^2
Since the mass cancels out, we can solve for the velocity:
Velocity at 53 ¢ª angle = sqrt((2.6 kg × 9.8 m/s^2 × 4.5 m) / 0.5)
(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. At the highest point of the swing, the total mechanical energy (sum of potential and kinetic energies) is equal to the total mechanical energy at the lowest point.
Potential Energy at the highest point + Kinetic Energy at the highest point = Potential Energy at the lowest point + Kinetic Energy at the lowest point
Using the formula for potential energy at any height:
Potential Energy = mass × acceleration due to gravity × height
The potential energy at the highest point is equal to the potential energy at the lowest point:
Mass × acceleration due to gravity × height at highest point = Mass × acceleration due to gravity × height at lowest point
Since the mass and acceleration due to gravity are the same, we can solve for the height at the highest point.
Height at highest point = Height at lowest point
This means that the greatest angle with the vertical is equal to the angle the string makes with the vertical at the lowest point, which is 90 ¢ª.
(c) To find the total mechanical energy of the system, we need to sum the potential and kinetic energies at the lowest point.
Total Mechanical Energy = Potential Energy at the lowest point + Kinetic Energy at the lowest point
Using the formulas for potential and kinetic energies:
Total Mechanical Energy = (mass × acceleration due to gravity × height) + (0.5 × mass × velocity^2)
Total Mechanical Energy = (2.6 kg × 9.8 m/s^2 × 4.5 m) + (0.5 × 2.6 kg × 7.9 m/s)^2
Now, plug in the values and calculate the total mechanical energy.