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Chapter 14: Lesson 1 - Periodic Motion and Potential Energy
Chapter 14: Lesson 1 - Periodic Motion and Potential Energy
Equations
Equations
It will be helpful to have your equation sheet ready. A copy of the equation sheet is under resource materials on this website and on google classroom. It can also be found HERE.
Periodic Motion
Periodic Motion
Periodic motion is any motion that repeats itself in a regular cycle.
Objects that move back and forth over the same path, have an equilibrium position. An equilibrium position is the position that the object is in when it is at rest. Some examples of objects that move back and forth through an equilibrium position are a playground swing, a vibrating guitar string, and an object on a string. When an object is in equilibrium, the net force is zero.
Whenever the object gets pulled away from its equilibrium position, the force in the system pulls it back toward equilibrium. This force could be gravity, as in the case of the swing or a pendulum, or a mass, as in the case below.
Hooke's Law
Hooke's Law
Hooke's Law states that the force exerted by a spring is directly proportional to the amount that the spring is stretched. The distance that a spring is stretched from the equilibrium point is abbreviated with a lowercase x. The distance that a spring is stretched, x, is measured in m.
Hooke's Law can also be expressed as an equation, F=kx, where F is the force on the spring, x, is the distance that the spring stretches from the equilibrium point, and k is the spring constant. If the spring is compressed, x is negative. If the spring is stretched, x is positive.
The spring constant tells us how springy or stiff a spring is. It is a constant for each spring. The units for k are N/m.
When a force is applied to stretch a spring, such as by hanging a mass at the end, like the previous picture, there is a direct linear relationship between the exerted force and the displacement. The more force you have pulling on the spring, the more the spring is stretched.
The slope of this graph is equal to the spring constant, which has units of Newtons per meter.
The more stiff a spring is (the harder it is to compress or stretch) the steeper the line. The more springy a spring is (the easier it is to compress or stretch) the less steep the line is.
The area under the curve represents the work done to stretch the spring, and therefore equals the elastic potential energy that is stored in the spring as a result of that work. The base of the triangle is x, and the height is the force, which (according to Hooke's Law) is equal to kx.
The potential energy in a spring is equal to one-half time the product of the spring constant and the square of the displacement. The unit for potential energy in a spring is Nm or Joules.
When an object hangs on a spring, the spring stretches until its upward force, Fsp, balances the object's weight, Fg.
When this occurs, the object is in equilibrium.
If you pull the object down, the spring force increases producing a net force upward. When you let go of the object, it accelerates in the upward direction. As the stretch of the spring is reduced, the upward force decreases.
The upward force of the spring and the object's weight are equal, there is no acceleration. The net force is zero and the object is in equilibrium.
Because there is no net force, the object continues its upward velocity, moving above the equilibrium position.
When the net force is in the direction opposite the displacement of the object and is directly proportional to the displacement, the object moves with simple harmonic motion.
Simple harmonic motion is a repeated back and forth movement over the same path about an equilibrium position, such as a mass on a spring or pendulum.
Pendulums
Pendulums
Simple harmonic motion can be demonstrated by the swing of a pendulum. A simple pendulum consists of a massive object called a bob, suspended by a string or light rod of length l. After the bob is pulled to one side and released, it swings back and forth.
The string or rod exerts a tension force, FT, and gravity exerts a force, Fg, on the bob. The vector sum of the two forces produces the net force, as shown in the picture to the left.
At the left and right positions, the net force and acceleration are at a maximum and the velocity is zero.
At the middle position (the equilibrium), the net force and acceleration are zero and the velocity is at a maximum.
The net force is a restoring force, it is in the opposite direction of the displacement of the bob and is trying to restore the bob to its equilibrium position.
Period of a Pendulum
Period of a Pendulum
Remember that period, T, is the time that it takes for one complete revolution. In the case of a pendulum, it is one complete back and forth motion. We measure period in seconds.
The period of a simple pendulum of length, lowercase L, is shown by the equation.
The period depends only upon the length of the pendulum and the acceleration due to gravity, not on the mass of the bob or the amplitude of the oscillation.
Pendulums!
Pendulums!
Watch this video to see the different between a long and a short pendulum.
You have finished Lesson 1!
You have finished Lesson 1!
Be sure to head over to google classroom and fill out the exit pass.
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