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Simple Harmonic Motion

- Holt Physics
- Pages 438 - 451

Distinguish simple harmonic motion from other

forms of periodic motion.

- Periodic motion is motion in which a body moves

repeatedly over the same path in equal time

intervals. - Examples uniform circular motion and simple

harmonic motion.

Contd

- Simple Harmonic Motion (SHM) is a special type of

periodic motion in which an object moves back and

forth, along a straight line or arc. - Examples pendulum, swings, vibrating spring,

piston in an engine. - In SHM, we ignore the effects of friction.
- Friction damps or slows down the motion of the

particles. If we included the affect of friction

then its called damped harmonic motion.

Contd

- For instance a person oscillating on a bungee

cord would experience damped harmonic motion.

Over time the amplitude of the oscillation

changes due to the energy lost to friction. - http//departments.weber.edu/physics/amiri/directo

r/DCRfiles/Energy/bungee4s.dcr

State the conditions necessary for simple

harmonic motion.

- A spring wants to stay at its equilibrium or

resting position. - However, if a distorting force pulls down on the

spring (when hanging an object from the spring,

the distorting force is the weight of the

object), the spring stretches to a point below

the equilibrium position. - The spring then creates a restoring force, which

tries to bring the spring back to the equilibrium

position.

Contd

- The distorting force and the restoring force are

equal in magnitude and opposite in direction. - FNET and the acceleration are always directed

toward the equilibrium position.

Contd

- Applet showing the forces, displacement, and

velocity of an object oscillating on a spring. - https//ngsir.netfirms.com/englishhtm/SpringSHM.ht

m

Displacement Velocity Acceleration

Contd

- at equilibrium
- speed or velocity is at a maximum
- displacement (x) is zero
- acceleration is zero
- FNET is zero (magnitude of Restoring Force

magnitude of Distorting Force which is the

weight) - object continues to move due to inertia

Contd

- at endpoints
- speed or velocity is zero
- displacement (x) is at a maximum equal to the

amplitude - acceleration is at a maximum
- restoring force is at a maximum
- F -kx (hooks law)
- FNET is at a maximum

State Hookes law and apply it to the solution of

problems.

- Hookes Law relates the distorting force and the

restoring force of a spring to the displacement

from equilibrium.

Contd

- F restoring force in Newtons
- k spring constant or force constant (stiffness

of a spring) in Newtons per meter (N/m) - x displacement from equilibrium in meters
- The distorting force is equal in magnitude but

opposite in direction to the restoring force.

The spring shown to the right has an unstretched

length of 3 cm. When a 2 kg object is hung from

the spring, it comes to rest at the 7 cm mark.

What is the spring constant of this spring? 490

N/m What direction is the restoring force? upward

Calculate the frequency and period of any simple

harmonic motion.

- T period (time required for a complete

vibration) in seconds - f frequency in vibrations / second or Hertz

- A particle is moving in simple harmonic motion

with a frequency of 10 Hz. - What is its period?
- 0.1 sec
- How many complete oscillations does it make in

one minute? - 600 oscilations

Relate uniform circular motion to simple harmonic

motion.

- The reference circle relates uniform circular

motion to SHM. - The shadow of an object moving in uniform

circular motion acts like SHM. - The speed of an object moving in uniform circular

motion may be constant but the shadow wont move

at a constant speed. - The speed at the endpoints is zero and a maximum

in the middle. - The shadow only shows one component of the motion.

Contd

- Applet showing the forces, displacement, and

velocity of an object oscillating on a spring and

an object in uniform circular motion. - http//www.physics.uoguelph.ca/tutorials/shm/phase

0.html

Identify the positions of and calculate the

maximum velocity and maximum accelerations of a

particle in simple harmonic motion.

- The acceleration is a maximum at the endpoints

and zero at the midpoint. - The acceleration is directly proportional to the

displacement, x. - The radius of the reference circle is equal to

the amplitude. - The force and acceleration are always directed

toward the midpoint.

The mass on the end of a spring (which stretches

linearly) is in equilibrium as shown. It is

pulled down so that the pointer is opposite the

11 cm mark and then released. What is the

amplitude of the vibration? 4 cm What two places

will the restoring force be greatest? 11 cm and 3

cm Where will the restoring force be least? 7 cm

- Where is the speed greatest?
- 7 cm
- What two places is the speed least?
- 3 cm and 11 cm
- Where is the magnitude of the displacement

greatest? - 3 cm and 11 cm
- Where is the displacement least?
- 7 cm
- Where is the magnitude of the acceleration

greatest? - 3 cm and 11 cm
- Where is the acceleration least?
- 7 cm
- Where is the elastic potential energy the

greatest? - Where is the kinetic energy the greatest?

Contd

- Remember that in uniform circular motion, the

velocity is calculated using

In SHM, the maximum velocity would be equal to

the velocity of the object in uniform circular

motion. The radius of the circle correlates to

the Amplitude (A) in SHM.

Contd

- Remember that in uniform circular motion, the

centripetal acceleration is calculated using

In SHM, the maximum acceleration would be equal

to the acceleration of the object in uniform

circular motion. The radius of the circle

correlates to the Amplitude (A) in SHM.

- A mass hanging on a spring oscillates with an

amplitude of 10 cm and a period of 2 seconds.

What is the maximum speed of the object and where

does it occur? - 0.314 m/s at equilibrium
- What is the minimum speed of the object and where

does it occur? - 0 m/s at the end points.
- What is its maximum acceleration?
- 0.987 m/s2 at the end points

- An object moving is simple harmonic motion can be

located using - A is amplitude
- f is frequency
- x is displacement from equilibrium
- ? is angular velocity

- The mass on the end of a spring (which stretches

linearly) is in equilibrium as shown. - It is pulled down so that the pointer is

opposite the 11 cm mark and then released. A

spring vibrates in SHM according to the equation

x 4 cospt. - How many complete vibrations does it make in 10

seconds? - 5 vibrations

- The elastic potential energy content of the

system is

So the maximum elastic potential energy is stored

at the end points of the oscillations where the

displacement is equal to the amplitude of the

vibration

At the end point, the object is not moving so

there is no kinetic energy. Therefore the total

energy content of the system is equal to

- A mass on a spring oscillates horizontally on a

frictionless table with an amplitude of A. In

terms of Eo (total mechanical energy of the

system) when the mass is at A, Us ______ and K

_________. - Us Eo and K 0
- When the mass is at 0.5 A, then Us __________

and K _________. - Us 0.25 Eo and K 0.75Eo
- When the mass is at the equilibrium position,

then Us _________ and K ________ - Us 0 and K Eo

- A 2 kg object is attached to a spring of force

constant k 500 N / m. The spring is then

stretched 3 cm from the equilibrium position and

released. What is the maximum kinetic energy of

this system? - 0.225 J
- What is the maximum velocity it will attain?
- 0.47 m/s

Contd

- T period (s)
- m mass (kg)
- k spring constant (N/m)

- You want a mass that, when hung on the end of the

spring, oscillates with a period of 3 seconds.

If the spring constant is 5 N/m, the mass should

be _______. - 1.14 kg

- The period for a mass vibrating on very stiff

springs (large values of k) will be (larger /

smaller) compared to the same mass vibrating on a

less stiff spring. - Smaller
- If the value of k halves, the period will be

______ times as long.

Relate the motion of a simple pendulum to simple

harmonic motion.

- A pendulum is a type of SHM.
- A simple pendulum is a small, dense mass

suspended by a cord of negligible mass. - The period of the pendulum is directly

proportional to the square root of the length and

inversely proportional to the square root of the

acceleration due to gravity.

Contd

- T period (s)
- l length (m)
- g acceleration due to gravity (m/s2)

- A pendulum has a period of 2 seconds here on the

surface of the earth. That pendulum is taken to

the moon where the acceleration due to gravity is

1/6 as much. What is the period of the pendulum

on the moon? - Squareroot of 6 times as much or 4.9 seconds.