Pendulum - find maximum angular displacement

[Lunadora]

Homework Statement

A 15-centimeter pendulum moves according to the equation:

theta=0.2cos8t

where theta is the angular displacement from the vertical in radians and t is the time in seconds. Determine the maximum angular displacement and the rate of change of theta when t=3 seconds.

Homework Equations

See, here's where I get stuck. It doesn't seem like I'm given enough information to do ANYTHING with this problem. At first I thought I could find the absolute maximum value by solving for theta at the endpoints and critical numbers, but I don't have any endpoints. Any physics equations I could use go out the window as well, because I have no initial displacement or velocity or any such stuff.

The Attempt at a Solution

Insert an hour of frustrated grumbling here, with no results.

 

[Delphi51]

Graph y=0.2cos8t to see the whole picture!
Or recall what value(s) of x make cos x a maximum. Graph y = cos x if you don't recall.

 

[kerimek]

I understand your frustration with this problem. However, there are a few things we can do to solve it.

Firstly, we can use the given equation to plot a graph of theta versus time. This will give us a visual representation of the pendulum's motion and we can use it to estimate the maximum angular displacement.

Secondly, we can use the equation for simple harmonic motion, theta = A cos(wt), where A is the amplitude and w is the angular frequency, to find the maximum angular displacement. In this case, we know that A = 0.2 radians and w = 8 radians/second. So, the maximum angular displacement can be found by substituting these values into the equation: theta = 0.2 cos(8t). This will give us a maximum angular displacement of 0.2 radians.

Finally, to find the rate of change of theta at t = 3 seconds, we can differentiate the given equation with respect to time. This will give us the angular velocity, which is the rate of change of theta. So, we can use the equation: omega = -A w sin(wt), where omega is the angular velocity. Substituting the values of A and w, we get omega = -0.2*8*sin(8t). At t = 3 seconds, the angular velocity will be -0.2*8*sin(8*3) = -0.2*8*sin(24) = approximately -3.14 radians/second.

I hope this helps you solve the problem. Remember, as a scientist, it's important to think outside the box and use different approaches to solve problems. Good luck!