The periodic nature of SHM and simple pendulum
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- The spring of a 1200kg car compresses 2.0mm when its 75kg driver gets into the car. If the car goes over a bump, what will be the frequency of the vibration?
- A 6 kg block is attached to a spring wit a spring constant of 216 N/m. The spring is stretched to a length of 12cm and then released.
- A mass of 2.40 kg is attached to a horizontal spring with a spring constant of 121N/m. It is stretched to a length of 10.0cm and released from test.
- A mass is attached to a horizontal spring, and oscillates with a period of 1.4s and with an amplitude of 12cm. At t=0s, the mass is 12cm to the right of the equilibrium positon.
- A simple Pendulum has a length of 42.0cm and makes 62.0 complete oscillation in 3.0 min.
- The length of a simple pendulum in 0.86m, the pendulum bob has a mass of 265 g and it is released to an angle of 11.0° to the vertical.
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Topic Notes
In this lesson, we will learn:
- The nature of the periodic motion
- The graph analysis of the periodic motion
- Simple Pendulum
Notes:
- If each vibration (the back and forth motion) takes the same amount of time, then the motion is Periodic.
- To discuss the period motion we need to define the following terms;
- Cycle; a complete to-and-fro movement.
- Amplitude; maximum displacement, the greatest distance from the equilibrium position.
- Period; time taken for one complete oscillation.
- Frequency; number of oscillations in one second.
- The following equations represent the mathematical relationship between frequency and period of motion;
Position as a Function of Time
Since the motion is considered as a periodic motion, we would be able to plot position Vs. time graph.
Looking at the graph we can refer to it as the cosine function, since at t = 0 the position is maximum;
w= angular velocity
w=2πf=2π/T
- For one complete cycle,
Velocity as a Function of Time
- Velocity is defined as the derivative of position with respect to time,
Since the Velocity is the sine function of time, at t=0,V=0, therefore; the graph starts at zero.
Acceleration as a Function of Time
- Acceleration is defined as the derivative of velocity with respect to time,
Simple Pendulum
- A simple pendulum is a small mass attached to the end of a string.
- The pendulum swings back and forth, ignoring the air resistance, it resembles simple harmonic motion.
Let’s apply the simple harmonic oscillator to the case of the simple pendulum;
In the case of simple pendulum;
A; is the maximum angular displacement, θmax
x; is the angle the pendulum is at; the angle is measured from the equilibrium positon (the vertical position),θ.
- The restoring force is opposite to the displacement and is equal to the component of the weight;
- In this case, the motion is considered to be simple harmonic motion if the angle is less than 15°, for small angles, sinθ≈θ;
Form Hooke’s law;
We know from spring- mass system;
T=2πkm⇒T=2πmg/Lm⇒T=2πgL
and f= 2π1 LgK=Lmg
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