A pendulum is released from some height. When it reaches the bottom of its swing, its speed is 2.5 m/s. What is the initial height of the pendulum?
To find the initial height of the pendulum, we can use the principle of conservation of mechanical energy. The total mechanical energy of the pendulum remains constant throughout its motion.
The mechanical energy of a pendulum consists of two components: potential energy (due to its height) and kinetic energy (due to its motion). At the maximum height, when the pendulum is released, its kinetic energy is zero, and the entire mechanical energy is potential energy. At the bottom of the swing, its potential energy is zero, and the entire mechanical energy is kinetic energy.
The equation for conservation of mechanical energy is:
PE_initial + KE_initial = PE_final + KE_final
Since the potential energy is directly proportional to the height and the kinetic energy is directly proportional to the square of the speed, we can express the equation as:
mgh + 0.5mv^2 = 0 + 0.5m(2.5)^2
Here, m represents the mass of the pendulum, g is the acceleration due to gravity (approximately 9.8 m/s^2), h is the initial height, and v is the speed at the bottom of the swing.
We can simplify the equation and solve for h:
gh = 0.5(2.5)^2
h = (0.5(2.5)^2) / g
By plugging in the values for g and solving the equation, we can find the initial height of the pendulum.