A pendulum has a mass of 3-kg, a length of 2.2 meter, and swings through a (half) arc of 30 degrees. To the tenth of a Joule, what is its maximum Kinetic Energy?
I just need a very simple, straightforward method of solving this problem. Unfortunately my textbook does a crappy job of explaining this. Thanks
Its maximum potential energy, refereed to the lowest (maximum kinetic energy) location, is
m g L(1 - cos 30)= 3*(9.8)*(2.2)*0.13397
= 8.7 J
That equals the maximum KE
To find the maximum kinetic energy of a pendulum, you can use the equation:
Kinetic Energy = 1/2 * mass * velocity^2
In this case, we need to find the velocity of the pendulum at its maximum swing.
To find the velocity, we can use the equation for the velocity of a pendulum:
velocity = √(2 * gravity * length * (1 - cos θ))
Where:
- gravity is the acceleration due to gravity (~9.8 m/s^2)
- length is the length of the pendulum (2.2 m)
- θ is the angle of the arc (30 degrees converted to radians)
First, let's convert the angle from degrees to radians:
θ = 30° * (π/180) ≈ 0.524 radians
Now, we can substitute the given values into the equation:
velocity = √(2 * 9.8 * 2.2 * (1 - cos 0.524))
Simplifying:
velocity ≈ √(2 * 9.8 * 2.2 * (1 - 0.866))
velocity ≈ √(2 * 9.8 * 2.2 * 0.134)
velocity ≈ √(5.6448)
velocity ≈ 2.377 m/s (rounded to three decimal places)
Finally, we can calculate the maximum kinetic energy using the formula:
Kinetic Energy = 1/2 * mass * velocity^2
Kinetic Energy = 1/2 * 3 kg * (2.377 m/s)^2
Kinetic Energy ≈ 8.481 J (rounded to one decimal place)
Therefore, the maximum kinetic energy of the pendulum is approximately 8.5 Joules.