Representing and Analyzing SHM
Đã giới thiệuSHM draws a perfect sine wave in time
- Track an oscillating mass and plot its position against time — you get a smooth, repeating wave.
- Not a jagged zig-zag, but the graceful curve of a sine or cosine.
- That shape is the fingerprint of simple harmonic motion, and it lets us predict every instant.
- Reading it tells you position, velocity and acceleration at a glance.
The equation of motion
- Starting from the maximum, position follows $x(t) = A\cos(\omega t)$.
- $A$ is the amplitude (the maximum displacement) and $\omega = 2\pi f$ is the angular frequency.
- After one period $T$ the curve repeats exactly.
- Feed in a time and the formula gives the exact position.
Position around a cycle
Step through the phase of the oscillation and see how position and energy vary across a cycle.
An oscillator is $x(t) = 0.1\cos(4t)$ (metres). What is its position at $t = 0$, in metres?
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$x(0) = 0.1\cos 0 = 0.1\ \text{m}$ — the maximum displacement (amplitude).
In $x(t) = A\cos(\omega t)$, the symbol $A$ is the ____.
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$A$ is the amplitude — the maximum displacement from equilibrium.
Where it's fastest and where it accelerates
- Speed is greatest at the middle (equilibrium) and zero at the ends (turning points).
- Acceleration is greatest at the ends and zero at the middle.
- This is because acceleration follows the force: $a = -\dfrac{k}{m}x$, biggest where $x$ is biggest.
- So velocity and acceleration are "out of step" — one peaks where the other is zero.
Where is an SHM oscillator moving fastest?
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Speed is greatest at the middle, where the acceleration (and force) is zero.
Where is the acceleration of an SHM oscillator greatest?
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$a = -\tfrac{k}{m}x$ is largest where $x$ is largest — at the extremes.
A mass starts at the far right. Order what happens next, first at the top.
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It accelerates toward the middle (max speed there), then decelerates to a stop at the other extreme.
Position, velocity, acceleration
- If position is a cosine, velocity is a (negative) sine, and acceleration is a (negative) cosine.
- Each is shifted a quarter-cycle from the last.
- The acceleration is always opposite the displacement — that is the SHM condition again.
- Together the three graphs tell the whole story of the motion.
In SHM the acceleration always points opposite to the displacement.
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$a = -\tfrac{k}{m}x$: the minus sign means acceleration opposes displacement.
In SHM, velocity and acceleration do not peak at the same place. Velocity is largest at the centre (where acceleration is zero); acceleration is largest at the extremes (where velocity is zero). Mixing these up is a classic error.
An oscillator has amplitude $A = 0.1\ \text{m}$ and angular frequency $\omega = 4\ \tfrac{\text{rad}}{\text{s}}$, starting at the maximum.
- Position: $x(t) = 0.1\cos(4t)$.
- At $t = 0$: $x = 0.1\cos 0 = 0.1\ \text{m}$ (the maximum), where the speed is zero.
SHM position follows $x(t) = A\cos(\omega t)$ — a sinusoid of amplitude $A$ and angular frequency $\omega = 2\pi f$. Speed peaks at the centre, acceleration peaks at the extremes, and acceleration always points opposite the displacement.
Write from a position of 0.3 m at the start
- Position is 0.3 times the cosine of 2 t, in metres.
- Find the position at t = 0 and say where the speed is greatest.
An oscillator has position 0.3 times the cosine of 2 t, in metres when t is in seconds. What is its position at t = 0, and where is it moving fastest?
Cosine of 0 is 1, so the position is 0.3 m.
What is the position at t = 0?
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At t = 0 the cosine is 1, so the displacement is the amplitude 0.3 m. Other options — 0.1 m (the lesson start); 0 (the middle of the motion).