Bell Work – Fill in the Chart Location MassGravityWeight Earth lbs Outer space 0 Earth's moon 0.17 Venus 0.90 Mars 0.38 Mercury 0.38 Jupiter 2.36 Saturn 0.92 Uranus 0.89 Neptune 1.13

Physical Science – Lecture 28 Simple Harmonic Oscillation

Oscillation Repetitive motion along the same path.

Common Features of Oscillation existence of an equilibrium a restoring force which grows stronger the further the system moves from equilibrium.

Examples Familiar examples include a swinging pendulum and AC power. Oscillations occur not only in physical systems but also in biological systems and in human society.

Simple Harmonic Oscillator The simplest mechanical oscillating system is a mass attached to a linear spring subject to no other forces. The other end of the spring is connected to a rigid support such as a wall.

Simple Harmonic Oscillator The system is in an equilibrium state when the spring is static. If the mass is moved from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium.

Simple Harmonic Oscillator

Momentum’s Effect In moving the mass back to the equilibrium position, it acquires momentum which keeps it moving beyond that position, establishing a new restoring force in the opposite sense.

Example

Kinetic and Potential Energy At the ends of the spring, the mass has potential energy stored in the spring. However, that potential energy is converted to kinetic energy to help restore the spring to equilibrium.

Calculating Net Force If the system is left at rest at the equilibrium position then there is no net force acting on the mass. When the mass is moved from the equilibrium position, a restoring force is exerted by the spring.

Calculating Restoring Force Equation F=kd

The Variables F is the restoring elastic force exerted by the spring (N), k is the spring constant (N/m) d is the displacement from the equilibrium position (in m).

How does it Restore? Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at x = 0, the momentum of the mass does not vanish due to the impulse of the restoring force that has acted on it. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then tends to slow it down, until its velocity vanishes, whereby it will attempt to reach equilibrium position again As long as the system has no energy loss, the mass will continue to oscillate. Thus, simple harmonic motion is a type of periodic motion.

Oscillator Position, velocity, and acceleration

The Period The time taken for an oscillation to occur is often referred to as the oscillatory period.

Another Equation

Rearranged Equation k = (4mπ 2 )/(T 2 )

The Variables T = period (seconds) π = “pie” – button on calculator m = mass (kilograms) k = spring constant (N/m)

Frequency of Simple Harmonic Motion

Effects of the Equation The equation shows that the period of oscillation is independent of both the amplitude and gravitational acceleration. This equation demonstrates that the simple harmonic motion is isochronous.

Putting it in the Calculator Scientific Calculator = 2, π, square root key, open parenthesis, value for m, divide, value for k, closed parenthesis, equals. Classroom Calculator = value for m, divide, value for k, =, square root key, x, 3.14, x, 2.

Practice Problem A 0.50kg object vibrates at the end of a vertical spring (k= 82N/m). What is the period of its vibration?

Practice Problem A block of mass m = 750 kg is attached to a wall by a spring with a spring constants of k = 1 N/m. Calculate the oscillation period.

Practice Problem A block of mass m = 160 kg is attached to a wall by a spring with a spring constants of k = 4 N/m. Calculate the oscillation period.

Practice Problem What force will an object with a spring constant of N/m require to return to equilibrium if the spring has been stretched 5 meters?

Practice Problem If a force of 30N is applied to a spring with a k value of N/m, how far will the spring stretch?

Practice Problem A 75 kg object vibrates at the end of a horizontal spring along a frictionless surface. If the period of vibration is 1.1 s, what is the spring constant?

Practice Problem After a bungee jump a 75kg student bobs up and down at the end of the bungee cord with a period of 0.23 s. What is the spring constant of the cord?

Practice Problem When a family of four (total mass of 200 kg) stepped into their 1200 kg car, the car’s springs compressed m. What is the spring constant of the car’s springs assuming they act as one single spring?

Practice Problem A 1000 kg car bounces up and down on its springs once every 2.0 s. What is the spring constant of its springs?