A 3.1 kg pendulum bob hangs from a 2 meter string. It is pulled to the side, position A, so that the bob is 98 cm (h on the sketch) above the lowest position, position C. The bob is released and begins to swing.
What is its potential energy after it has swung to position B, 34.0 cm below the starting point?
And, how fast is it moving at that point?
Its moving really fast
To determine the potential energy of the pendulum bob at position B, we can use the equation:
Potential energy = mass * acceleration due to gravity * height
First, let's calculate the potential energy at position A using the given data. The height at position A is 98 cm, which is equivalent to 0.98 meters. The mass of the pendulum bob is 3.1 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
Potential energy at position A = 3.1 kg * 9.8 m/s^2 * 0.98 m
= 29.82 Joules
Now, let's determine the potential energy at position B. The height at position B is given as 34 cm, which is equal to 0.34 meters.
Potential energy at position B = 3.1 kg * 9.8 m/s^2 * 0.34 m
= 10.246 Joules
So, the potential energy of the pendulum bob at position B is approximately 10.246 Joules.
To find the speed at position B, we can use the conservation of mechanical energy principle. The total mechanical energy of the pendulum is the sum of its potential energy and its kinetic energy.
Mechanical energy = Potential energy + Kinetic energy
At position A, all the energy is in the form of potential energy, as the bob is at its highest point. At position B, some potential energy is converted to kinetic energy as the bob moves downward.
Since mechanical energy is conserved, we can equate the mechanical energy at position A to the mechanical energy at position B.
Potential energy at position A = potential energy at position B + kinetic energy at position B
29.82 Joules = 10.246 Joules + kinetic energy at position B
Rearranging the equation, we can solve for the kinetic energy at position B.
Kinetic energy at position B = 29.82 Joules - 10.246 Joules
= 19.574 Joules
Finally, to find the speed at position B, we can use the equation:
Kinetic energy = 0.5 * mass * velocity^2
19.574 Joules = 0.5 * 3.1 kg * velocity^2
Rearranging the equation, we can solve for the velocity at position B.
velocity^2 = (19.574 Joules) / (0.5 * 3.1 kg)
velocity^2 = 12.560645161290322 m^2/s^2
Taking the square root of both sides, we can find the velocity at position B.
velocity = √(12.560645161290322 m^2/s^2)
velocity = 3.54 m/s
So, the pendulum bob is moving at a speed of approximately 3.54 m/s when it reaches position B.