1. PENDULUM
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2. SPRING
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3. SIMPLE HARMONIC
MOTION
SIMPLE HARMONIC
MOTION
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4. SIMPLE HARMONIC MOTION
ALL INVOLVE
SIMPLE HARMONIC MOTION
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5. A body will undergo SIMPLE HARMONIC MOTION when the force that tries to restore the object to its REST POSITION is PROPORTIONAL TO the DISPLACEMENT of the object.
A pendulum and a mass on a spring both undergo this type of motion which can be described by a SINE WAVE or a COSINE WAVE depending upon the start position.
Displacement x
+ A
- A
Time t
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6. DEFINITION
SHM is a particle motion with an acceleration (a) that is directly proportional to the particle’s displacement (x) from a fixed point (rest point), and this acceleration always points towards the fixed point.
Rest point A A x x or
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7. MORE DEFINITIONS T + A - A
Displacement x
time T
+ A
- A
Amplitude ( A ): The maximum distance that an object moves from its rest position. x = A and x = - A .
Period ( T ): The time that it takes to execute one complete cycle of its motion.
Units seconds,
Frequency ( f ): The number or oscillations the object completes per unit time.
Units Hz = s-1 .
Angular Frequency ( ω ): The frequency in radians per second, 2π per cycle.
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8. RADIANS IN RADIANS FOR A FULL CIRCLE RADIANS θ r Arc length s
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9. EQUATION OF SHM Acceleration – Displacement graph Gradient = - ω2x a
Gradient = - ω2
- A
+ A
MAXIMUM ACCELERATION = ± ω2 A = ( 2πf )2 A
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10. EQUATION FOR VARIATION OF VELOCITY WITH DISPLACEMENT+x -x x v
Maximum velocity, v = ± 2 π f A
Maximum Kinetic Energy, EK = ½ mv2 = ½ m ( 2 π f A )2
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11. Velocity = gradient of displacement- time graph
Displacement x t
Velocity v
Velocity = gradient of displacement- time graph
Maximum velocity in the middle of the motion t
ZERO velocity at the end of the motion
Acceleration = gradient of velocity - time graph
Acceleration a
Maximum acceleration at the end of the motion – where the restoring force is greatest! t
ZERO acceleration in the middle of the motion!
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12. l THE PENDULUM The period, T, is the time for one complete cycle.
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13. MASS ON A SPRING k = the spring constant in N m-1 M F = Mg = ke e A
Stretch & Release A
k = the spring constant in N m-1
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14. The link below enables you to look at the factors that influence the period of a pendulum and the period of a mass on a spring
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15. If damping is negligible, the total energy will be constant
ENERGY IN SHM
PENDULUM
SPRING M
potential
kinetic
potential EP
Potential EP
Kinetic EK
potential
If damping is negligible, the total energy will be constant
ETOTAL = Ep + EK
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16. Energy in SHM Maximum velocity, v = ± 2 π f A
Maximum Kinetic Energy, EK = ½ m ( 2 π f A )2 = 2π2 m f2 A2
Hence TOTAL ENERGY = 2π2 m f2 A2
= MAXIMUM POTENTIAL ENERGY!
For a spring,
energy stored = ½ Fx = ½ kx2,
[as F=kx] m
x = A m
x = 0 F
MAXIMUM POTENTIAL ENERGY = TOTAL ENERGY = ½ kA2
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17. Energy Change with POSITION
Energy in SHM
= kinetic
= potential
= TOTAL ENERGY, E
Energy Change with POSITION x -A +A
energy E
Energy Change with TIME
energy
time E
N.B. Both the kinetic and the potential energies reach a maximum TWICE in on cycle.
T/2 T
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18. DAMPING DISPLACEMENT INITIAL AMPLITUDE time
THE AMPLITUDE DECAYS EXPONENTIALLY WITH TIME
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