What is the acceleration of a block in simple harmonic oscillation at t=3.0s?

In summary, the conversation discusses a problem involving a block attached to a spring and its acceleration at a given time. Different equations and initial conditions are considered to find the acceleration, with a suggestion to check the calculator's setting for radians. The conversation ends with the participant thanking for the help and sharing a solution to the problem.
  • #1
fstam2
10
0
Simple Harmonic Oscillation

Hello, I am new here and wish I had found this forum earlier in the semester. Here is the situation:
A block (mass m=0.75 kg) rests on a horizontal surface (frictionless), attached to a horizontal spring (k=235 N/m). At time t=0, the block is located at the equilibrium position (x=0), given a sove that compresses the spring. The block gains a speed of 2.5 m/s instantaneously.
(ii) What is the acceleration of the block at time, t=3.0 s?

The equation I have found that might be correct is:
a= -(kA/m)sin(2(pie)t/T), where t= 3.0s and T=.352 based on
T= 2(pie)sqrt(m/k)
Using this equation I arrived at -35.2 m/s2 (the negative is because of the direction of the acceleration at t= 3.0s). This answer makes sense in that the acceleration has slowed from a max acceleration of 43.8 m/s2.

Any suggestions, advice, criticisms?
Thanks for your help.
Todd
 
Last edited:
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  • #2
Your equations are true, but I don't see how you got your answer. Like most problems, there's more than one way to do this, but considering the title of your post, consider one of the general forms for SHM of an object:

[itex] x(t) = A\sin(\omega_n t) + B\cos(\omega_n t) [/itex]

where x is the position at time t, ωn is the natural frequency, and A and B are constants determined by the initial conditions. The natural frequency in this problem is:

[itex] \omega_n = \sqrt{\frac{k}{m}} [/itex]

Using the fact that x(0) = 0, we find that B = 0.

The other initial condition is the initial velocity:

[itex] \dot{x}(0) = \omega_n A \cos(\omega_n * (0)) = -2.5\,{\rm \frac{m}{s}} [/itex]

(I have defined compression to be the negative x direction.)

Use this equation (with ωn = 17.7 rad/s) to find A = -0.141 m negative just means it starts moving in compression). You can write the acceleration from the SHM equation as well (by differentiating twice):

[itex] \ddot{x}(t) = -\omega_n^2 A\sin(\omega_n t) [/itex]

Solving for t = 3 s, the acceleration should be 13.2 m/s/s. You should rework the problem to see if I made a mistake, but I think I found your problem: is your calculator in degrees or radians right now? What should it be in? (I wish I noticed that before I blah-blahed the above explanation.)
 
  • #3
Thanks for your help. I only needed to change my calc. to radians to work out the problem. I never had the manual for the calc., and HP only has the Spanish version to download.
Todd
 

1. What is Simple Harmonic Oscillation?

Simple Harmonic Oscillation (SHO) is a type of periodic motion where an object oscillates back and forth around a central equilibrium point due to a restoring force that is directly proportional to the displacement from the equilibrium point.

2. What are the factors that affect Simple Harmonic Oscillation?

The two main factors that affect Simple Harmonic Oscillation are the mass of the object and the stiffness of the restoring force. The greater the mass, the longer it takes to complete one oscillation. Similarly, the stiffer the restoring force, the faster the oscillation will be.

3. What is the equation for Simple Harmonic Oscillation?

The equation for Simple Harmonic Oscillation is x = A sin(ωt + φ), where x is the displacement from equilibrium, A is the amplitude, ω is the angular frequency, and φ is the phase angle.

4. How is Simple Harmonic Oscillation different from other types of oscillations?

Simple Harmonic Oscillation is different from other types of oscillations because it follows a specific pattern of motion, where the displacement from equilibrium is directly proportional to the restoring force. Other types of oscillations may not follow this pattern and can have varying amplitudes and frequencies.

5. What are some real-life examples of Simple Harmonic Oscillation?

Some real-life examples of Simple Harmonic Oscillation include a pendulum, a mass-spring system, a swinging door, and a vibrating guitar string. These systems all exhibit SHO because they have a restoring force that is proportional to the displacement from equilibrium.

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