Simple Harmonic Motion Problem

In summary, the conversation discusses an object oscillating in simple harmonic motion at the end of a vertical spring. The equation for its vertical position as a function of time is given, and the conversation addresses questions about the spring constant, maximum acceleration, maximum speed, and time taken for the object to move from its highest to lowest point. The solution involves using the general equation for position in simple harmonic motion and the relationship between angular frequency and the spring constant.
  • #1
Stendhal
24
1

Homework Statement



A 12.0-N object is oscillating in simple harmonic motion at the end of an ideal vertical spring. Its vertical position y as a function of time t is given by:

y(t)=4.50cmcos[(19.5s−1)t−π/8].

(a) What is the spring constant of the spring?

(b) What is the maximum acceleration of the object?

(c) What is the maximum speed that the object reaches?

(d) How long does it take the object to go from its highest point to its lowest point?

Homework Equations


F=-kx

a=kx/m

The Attempt at a Solution


I am unsure as to where I need to start for part a.

12N=-k(.045m) since the amplitude is the max displacement of the spring.

12N/.045m = k ==> 267N/m. However, this is wrong.

I think I'm just missing something very obvious. Any help is very appreciated!
 
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  • #2
I would begin with comparing the general equation for position with this equation

[tex] y(t)=4.50 cos(19.5t−\dfrac{\pi}{8}) [/tex]
[tex] y(t)=A cos(\omega t−\varphi)[/tex]

We also know that [tex] \omega = \sqrt{\dfrac{k}{m}} [/tex] where k is the spring constant.
 
  • #3
Totally forgot about the general equation that describes harmonics. Whoops.

Well, after you told me that, I figured everything out on my own. Thank you!
 

1. What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion is a type of periodic motion in which a system oscillates back and forth around an equilibrium point, with the motion being proportional to the displacement from the equilibrium point and the acceleration always directed towards the equilibrium point.

2. What causes Simple Harmonic Motion?

Simple Harmonic Motion is caused by a restoring force that acts on a system, pulling it back towards the equilibrium point. This restoring force is proportional to the displacement from the equilibrium point and is what allows the system to oscillate in a periodic manner.

3. How is Simple Harmonic Motion represented mathematically?

The mathematical representation of Simple Harmonic Motion is given by the equation x(t) = A*sin(ωt + φ), where x is the displacement from the equilibrium point, A is the amplitude, ω is the angular frequency, and φ is the phase constant. This equation can also be written in terms of velocity and acceleration as v(t) = ω*A*cos(ωt + φ) and a(t) = -ω^2*A*sin(ωt + φ).

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

Some common examples of Simple Harmonic Motion include the motion of a pendulum, the vibrations of a guitar string, and the oscillations of a mass-spring system. Other examples include the motion of a swing, the motion of a mass attached to a vertical spring, and the motion of a mass attached to a horizontal spring and pulled by a constant force.

5. How do you solve problems involving Simple Harmonic Motion?

To solve problems involving Simple Harmonic Motion, you can use the equations mentioned in question 3 along with the given information about the system (such as the amplitude, frequency, and initial conditions) to find the displacement, velocity, and acceleration of the system at any given time. It is also important to understand the concept of energy conservation in Simple Harmonic Motion and how it relates to the equations of motion.

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