Simple Harmonic Motion

Periodic Motion

Acrobat on a trapeze Child on a swing Pendulum of a clock Mass attached to a spring

When the spring is stretched to the right, the spring force pulls the mass to the left.

When the spring is unstretched, the spring force is zero.

When the spring is compressed to the left, the spring force is directed to the right.

The direction of the force acting on the mass is always opposite the direction of the mass’s displacement from equilibrium (x=0).

At the equilibrium point, the velocity reaches a maximum…(think about projectile motion…)

The force that is pulling the spring back towards the equilibrium point is called the restoring force.

Any periodic motion that is the result of a restoring force is called:

simple harmonic motion.

In 1678 Robert Hooke found that most mass- spring systems obey a simple relationship between force and displacement. For small displacements from the equilibrium:

Felastic =-kx

Felastic =-kx

The negative sign indicates that the direction of the spring force is always opposite the direction of the mass’s displacement from equilibrium

In other words… the negative sign shows that the spring force will tend to move the object back to its equilibrium position.

Felastic =-kx

Spring force = -(spring constant x displacement)

SI units of ‘k’ are N/m

‘k’ is the spring constant. The value of ‘k’ is different for each individual spring (much like μ is unique for two different surfaces).

A greater value of ‘k’ means a stiffer spring… b/c a greater force is needed to stretch or compress the spring.

A mass of .55 kg attached to a vertical spring is stretched with an additional force of 2N. The spring 2.0 cm from its original equilibrium position. What is the spring constant?

A slingshot consists of a light leather cup attached to two rubber bands If it takes a force of 32N to stretch the bands 1.2 cm, what is the equivalent spring constant of the rubber bands?

Stretched and compressed springs contain elastic potential energy.

When a spring is held in a position that is completely stretched ( or completely stretched) the kinetic energy is converted to elastic potential energy

Can you make the connection to elastic potential energy and potential energy?

The Simple Pendulum

The simple Pendulum will have an increase in gravitational potential energy as the displacement increases.

The restoring force on a pendulum bob is a component of the bob’s weight it is the component of the force that is

perpendicular to the string

The maximum displacement from equilibrium position is called the amplitude.

Amplitude is typically measured as the degree or the angle between the pendulum’s equilibrium position and its max displacement.

The period is the time it takes for one complete cycle of motion for the pendulum. Notice that after the time period ‘T’ the object will be back where it started.

The frequency is the inverse of the period…

Frequency is the number of cycles per unit time.

(The unit of frequency is a Hertz)

What influences the time it takes for a pendulum to swing?

The formula for the period of a simple pendulum in simple harmonic motion:

So, the question is:

If I have a pendulum of length “L,” how does adding mass to the end influence the period?

Calculate the period and frequency of a 3.50 m long pendulum at the following locations:

The North Pole, where g = 9.832 m/s2

Chicago, where g= 9.803 m/s2

Jakarta, Indonesia, where g = 9.782 m/s2

A trapeze artist swings in simple harmonic motion with a period of 3.8 s. Calculate the length of the cables supporting the trapeze.

So – now we can calculate the period of a mass-spring system…

T = 2π√m/k

A mass of .30 kg is attached to a spring and is set into vibration with a period of .24 s. What is the spring constant of the spring?

When a mass of 25 g is attached to a certain spring, it makes 20 complete vibrations in 4.0s. What is the spring constant in the spring?

A spring of spring constant 30.0 N/m is attached to different masses. And the system is set in motion. Find the period and frequency of vibration for masses of the following magnitudes: 2.3 kg

15 g

1.9 kg