show that mgh = 1/2mv^2 for a simple pendulum
length L
angle from vertical = theta
height above bottom = L * (1 - cos theta)
for small angles cos theta = 1 - theta^2/2
so for small A height = L * theta^2/2
so potential energy at top = m g L * theta^2/2
if max theta = a then max pe = m g L a^2/2 =( ma^2/2) g L
if theta = a sin 2 pi f t
omega =d theta/dt = a (2 pi f) cos wt
max omega = a (2 pi f) when t = 0 at bottom
max velocity = L * a * 2 pi f
max Ke = (1/2) m ( L * a * 2 pi f)^2 = (ma^2 / 2) L^2 (2pif)^2
now
does g L = L^2(2 pi f)^2
or (2 pi f)^2 = g/L ?????? I think so :)
To demonstrate that the expression mgh = 1/2mv² holds for a simple pendulum, we will make use of a few fundamental principles of physics.
Let's begin by considering the motion of a simple pendulum, which consists of a mass (m) attached to a string or rod of length (L). The pendulum is free to swing back and forth under the influence of gravity.
The potential energy (PE) of the pendulum is given by the height (h) of the mass above the equilibrium position. This potential energy is equal to the work done by gravity to lift the mass to that height. Thus, we have:
PE = mgh,
where g represents the acceleration due to gravity.
The kinetic energy (KE) of the pendulum is given by the motion of the mass. As the pendulum swings, the mass has a velocity (v), which is dependent on its position and the length of the pendulum.
The equation for the kinetic energy of an object is given by KE = 1/2mv², where v represents the velocity of the object.
At the highest point of the swing, when the mass momentarily stops and changes direction, all of its kinetic energy is converted into potential energy. This occurs because the velocity is zero at the highest point of the swing. At this same point, the potential energy is maximum, which means that the two energies are equal. Therefore, we can equate the two equations:
PE = KE,
mgh = 1/2mv².
Hence, we can conclude that mgh = 1/2mv² for a simple pendulum.
To show that there is a relationship between the potential energy (mgh) and the kinetic energy (1/2mv^2) of a simple pendulum, we can apply the principles of conservation of mechanical energy.
A simple pendulum consists of a mass (m) attached to a string or rod of length (L) that is free to swing back and forth. When the pendulum oscillates, it goes through points where it reaches its highest position (referred to as maximum displacement) and its lowest position (referred to as equilibrium position).
Let's consider the two extreme positions of the pendulum: the highest point and the lowest point.
At the highest point, the pendulum has maximum potential energy (mgh) and minimum kinetic energy (1/2mv^2). This is because it momentarily stops and changes its direction of motion.
At the lowest point (equilibrium position), the pendulum has minimum potential energy (zero) and maximum kinetic energy (1/2mv^2). This is because all the potential energy has been converted into kinetic energy as the pendulum reaches its maximum speed.
Conservation of mechanical energy states that the sum of potential energy and kinetic energy remains constant throughout the motion of the pendulum, assuming no external forces are acting on it (such as air resistance).
Therefore, we can state that the potential energy (mgh) at the highest point is equal to the kinetic energy (1/2mv^2) at the lowest point of the pendulum swing.
It's important to note that this relationship holds only for small angles in a simple pendulum, as the derivation is based on using small angle approximations.