# Simple Harmonic Motion position and velocity problem

• madchemist
In summary: I\sin(\omega t + \phi)In summary, at t=0 the particle is at its equilibrium position, and its velocity and acceleration are zero.
madchemist
"A particle moves along the x-axis. It is initially at the position 0.270 m, moving with velocity 0.140 m/s and acceleration -0.320 m/s^2. Assume it moves with simple harmonic motion for 4.50 s and x=0 is its equilibrium position. Find its position and velocity at the end of this time interval."

x=Amplitude * cos2(pi)ft

Found f using T= radical (x/a) * 2pi = 5.769 so f= 0.1734 Hz

However, even with solving for f I'm still left with two unknowns, i.e. A and t. Please help...

Last edited:
You are given data for the initial postion, velocity, and acceleration. Set up equations for each. That will allow you to solve for the amplitude and phase.

madchemist said:
"A particle moves along the x-axis. It is initially at the position 0.270 m, moving with velocity 0.140 m/s and acceleration -0.320 m/s^2. Assume it moves with simple harmonic motion for 4.50 s and x=0 is its equilibrium position. Find its position and velocity at the end of this time interval."

x=Amplitude * cos2(pi)ft

Found f using T= radical (x/a) * 2pi = 5.769 so f= 0.1734 Hz

However, even with solving for f I'm still left with two unknowns, i.e. A and t. Please help...

This is how I've done it quickly...

x(t) = Bsin[2pi.f.t] + Acos[2pi.f.t] NB because at t=0, x=0.27 you can ignore the sin part
x(t) = Acos[2pi.f.t]
v(t) = -A.2pi.f.sin[2pi.f.t] (Diffrentiated once x(t) with respect to t)
a(t) = -A.(2pi.f)^2.cos[2pi.f.t] (Diffrentiated twice x(t) with respect to t)

x(t=0) = Acos[2pi.f.0]
x(t=0) = A (cos(0) = 0)
A = 0.27

a(t=0) = -0.27.(2pi.f)^2.cos[2pi.f.0]
a(t=0) = -0.27.(2pi.f)^2 (cos(0) = 0)
-0.320 = -0.27.(2pi.f)^2
f = sqrt[0.320/(0.27.(2.pi)^2)]
f = 0.173

x(t) = 0.27.cos[2pi.0.173.t]
x(t=4.5) = 0.27.cos[2pi.0.173.4.5]
x(t=4.5) = 0.050m

Dont quote me though its a while since I've done SHM

Cynapse said:
This is how I've done it quickly...

x(t) = Bsin[2pi.f.t] + Acos[2pi.f.t] NB because at t=0, x=0.27 you can ignore the sin part
x(t) = Acos[2pi.f.t]
v(t) = -A.2pi.f.sin[2pi.f.t] (Diffrentiated once x(t) with respect to t)
a(t) = -A.(2pi.f)^2.cos[2pi.f.t] (Diffrentiated twice x(t) with respect to t)

x(t=0) = Acos[2pi.f.0]
x(t=0) = A (cos(0) = 0)
A = 0.27
Sanity check: Do you think that the particle is at maximum displacement at t = 0?

Better to start with this:
$$x = A\sin(\omega t + \phi)$$

## 1. What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion (SHM) is a type of periodic motion in which the restoring force is directly proportional to the displacement from equilibrium. This results in a sinusoidal pattern of motion.

## 2. What is the equation for SHM position and velocity?

The equation for SHM position is x = Acos(ωt + φ), where x is the position, A is the amplitude, ω is the angular frequency, and φ is the phase constant. The equation for SHM velocity is v = -ωAsin(ωt + φ).

## 3. How is SHM different from other types of motion?

SHM is different from other types of motion because it is characterized by a constant frequency and amplitude, and the motion is always directed towards the equilibrium position. Other types of motion may have varying frequencies and amplitudes, and may not necessarily have a restoring force towards equilibrium.

## 4. How do you solve a SHM position and velocity problem?

To solve a SHM position and velocity problem, you first need to identify the given values for amplitude, angular frequency, and phase constant. Then, plug these values into the appropriate equations (x = Acos(ωt + φ) for position and v = -ωAsin(ωt + φ) for velocity) to calculate the position and velocity at a given time.

## 5. What are some real-life examples of SHM?

Some real-life examples of SHM include the motion of a mass on a spring, a pendulum swinging back and forth, and the vibrations of a guitar string. SHM can also be seen in the motion of a swinging door, the bobbing of a buoy in water, and the movement of a diving board.

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