It is not obvious that a sinusoidally-varying displacement is a consequence of the condition that the acceleration is proportional to displacement but in the opposite direction!
However, we can verify that the condition for simple harmonic motion is obeyed by differentiating the original function for displacement twice with respect to time.
Key Concepts
By analysing the motion of a pendulum we find the displacement is sinusoidally related to time. This means that the displacement equation with respect to time will contain either sin or cos.
The velocity is found from the gradient of the displacement time graph. If, for example, the displacement is a sin curve then velocity will be a cos curve.
Acceleration is the gradient of the velocity time graph. Sticking with velocity as a cos curve, acceleration will be a -sin curve. This is proportional to but in the opposition direction to the displacement time curve.
Imagine watching circular motion from the side, with your eye on the same plane as the circle itself. You would see the object moving from side-to-side or up-and-down repeatedly. The equations of this motion are the same as simple harmonic motion.
This can be used to show that \(a=-\omega^2x\).
The equation for the displacement of a body executing SHM is either:
- \(x = x_o\sin(2πft)\)
- \(x = x_o\cos(2πft)\)
Use quizzes to practise application of theory.
START QUIZ!
The diagram represents 3 sin curves (horizontal axis is time)
If the green line is displacement then the blue line is
Motion represented is SHM so a is proportional to -x
The diagram represents 3 sin curves (horizontal axis is time)
If the green line is displacement the red line is
Velocity is the gradient of the dispacent- time graph. The red line is - gradient
This graph represents the displacement time graph for a body moving with SHM. The axis are in meters and seconds.
The maximum velocity is approximately
The gradient gives the velocity. This is certainly bigger than 1.5
A pendulum bob is released from the position shown
Which graph represents the displacement
left is negative
A pendulum bob is released from the position shown
Which graph represents the velocity
Starts from zero moving to the right (+)
A pendulum bob is released from the position shown
Which graph represents the acceleration
Starts max and positive direction
What is the equation of this line
just have to know
The equation for the displacment of a body executing SHM is of the form
y = A sin (2πBt)
B in this equation is
y = a sin (2πft)
The greph represents two oscillations
The equation of the black line is sin(2πft). The equation of the red line is.
The red line is π/2 in front of the black.
2π is one cycle so π/2 is 1/4 of a cycle
The diagram below represents a ball moving anticlockwise in a circle at a constant speed .
Which line represents the horizontal displacement if the clock is started when the ball is at the position shown.
Strats with zero horizontal displacement and goes left of centre.
The diagram below represents a ball moving anticlockwise in a circle at constant speed
If the graph repreents the vertical component of the motion where was the ball when time = 0s
Starts with zero vertical displacement and goes down.
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