A pendulum has a mass of 3-kg, a length of 1.3 meter, and swings through a (half) arc of 30 degrees. To the tenth of a Joule, what is its maximum Kinetic Energy?
Calculate the potential energy gain when swinging to maximum height.
m g L (1 - cos30)
so do you just plug everything into this equation and that's it? What exactly do is this expression for?
Yes, that is the answer. Plug in and get the number. It represents the difference between maximum and minimum potential energy, and that equals the maximum kinetic energy, since at maximum P.E. (top of swing), the K.E is zero.
To calculate the maximum kinetic energy of the pendulum, we need to use the formula for kinetic energy.
The formula for kinetic energy is given by:
K.E. = (1/2) * m * v^2,
where K.E. is the kinetic energy, m is the mass of the pendulum, and v represents the velocity of the pendulum.
First, let's calculate the velocity of the pendulum at its maximum swing.
The maximum potential energy (PE) of the pendulum is converted entirely to kinetic energy at the lowest point of its swing. Therefore, using the principle of conservation of energy, we can equate the potential energy at the highest point to the kinetic energy at the lowest point.
PE = m * g * h = K.E.
Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the height.
In this case, the height h can be calculated as the vertical distance from the highest to the lowest point of the pendulum's swing.
Given that the pendulum swings in a (half) arc of 30 degrees, we can use trigonometry to find the height:
h = l * (1 - cosθ),
where l is the length of the pendulum and θ is the arc angle.
Using the given values:
l = 1.3 meters
θ = 30 degrees
h = 1.3 * (1 - cos(30))
≈ 1.3 * (1 - 0.866)
≈ 1.3 * 0.134
≈ 0.1742 meters
Now, we can calculate the potential energy at the highest point:
PE = m * g * h
= 3 kg * 9.8 m/s^2 * 0.1742 meters
≈ 5.0992 Joules.
As per the principle of conservation of energy, this potential energy is converted entirely into kinetic energy when the pendulum reaches its lowest point. Hence, the maximum kinetic energy is also approximately equal to 5.0992 Joules.