Simple Harmonic Motion, velocity and acceleration

In summary, the problem involves finding the velocity and acceleration of a body undergoing simple harmonic motion along the x-axis, with its displacement being described by the equation x=5.0 sin (pi*(t) + pi/3). To find the velocity and acceleration, the equation for position, x(t), can be differentiated to get the equations for velocity, v(t), and acceleration, a(t). By plugging in t=1, the desired values can be obtained. Additionally, to convert between sine and cosine, the trigonometric identities \sin[\theta]=\cos\left[\frac{\pi}{2}-\theta\right] and \cos[\theta]=\sin\left[\frac{\pi}{2}-\
  • #1
hodgepodge
47
0

Homework Statement


a body oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x=5.0 sin (pi*(t) + pi/3). What is the velocity in m/s and acceleration in m/s^2 of the body at t=1.0s?


Homework Equations


x(t) = A cos (omega*(t) + phi)


The Attempt at a Solution


i am just confused as to how the equation in the problem is sin, vs. cos in the relevant equation, doesn't this just mean that the period of the motion of the object starts at a different point and can i use pi*(t) to find omega*(t) and pi/3 to find phi and get my velocity and accelerations, or do i have to manipulate the equation in the problem to get it in the form in relevant equations...if so what would i do to get the sin into cos?

thanks
 
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  • #2
Well you are given a position,

[tex]
x(t)=5.0\sin\left[\pi t+\frac{\pi}{3}\right]
[/tex]

and how do you get a velocity, [itex]v(t)[/itex], from a position? Similarly, how do you get an acceleration, [itex]a(t)[/itex], from a velocity?
 
  • #3
so i could just take the derivative and find instantaneous velocity and then take second derivative for instantaneous acceleration?
 
  • #4
Correct. After taking the derivatives, just put in for [itex]t=1[/itex] and you'll have your velocity and accelerations at the appropriate time.


Also, since I didn't quite answer Part 3 very well, to get cosine from sine (and vice versa):

[tex]
\sin[\theta]=\cos\left[\frac{\pi}{2}-\theta\right]
[/tex]

[tex]
\cos[\theta]=\sin\left[\frac{\pi}{2}-\theta\right]
[/tex]
 
  • #5
Thank you so much.
 

Related to Simple Harmonic Motion, velocity and acceleration

What is Simple Harmonic Motion?

Simple Harmonic Motion is a type of oscillatory motion where an object moves back and forth in a regular pattern. It is characterized by a restoring force that is proportional to the displacement from the equilibrium position.

What is the formula for calculating velocity in Simple Harmonic Motion?

The formula for calculating velocity in Simple Harmonic Motion is v = Aω cos(ωt), where A is the amplitude of the motion, ω is the angular frequency, and t is the time.

How does Simple Harmonic Motion relate to acceleration?

In Simple Harmonic Motion, acceleration is directly proportional to the displacement from the equilibrium position and is always directed towards the equilibrium point. This means that as the displacement increases, so does the acceleration, and vice versa.

What is the difference between Simple Harmonic Motion and Simple Harmonic Oscillation?

Simple Harmonic Motion refers to the back and forth movement of an object, while Simple Harmonic Oscillation refers to the entire process of the object moving back and forth, including the time it takes to complete one full cycle.

What are some real-life examples of Simple Harmonic Motion?

Some real-life examples of Simple Harmonic Motion include the back and forth movement of a swing, the motion of a mass on a spring, and the vibrations of a guitar string.

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