At what position is the kinetic energy of the pendulum at its maximum?

Correct answer:

Kinetic energy

Explanation:

For this question, we're presented with a scenario in which a pendulum is swinging back and forth. Thus, we know that this pendulum is an example of simple harmonic motion. As the pendulum swings back and forth, a number of its variables change in a cyclical fashion.

First, let's take a look at potential energy. When the pendulum is at the bottom of its trajectory, its potential energy will also be at its minimum. This is because the height of the mass attached to the pendulum is lowest at this point. We can show this with the following expression, where the 

Next, let's consider tangential acceleration. When the pendulum reaches its highest point, it will briefly be at rest for a very short instant. At this highest point, the pendulum also has its greatest amount of potential energy. When the pendulum begins to fall, the force due to gravity causes the pendulum to fall. However, it's important to realize that the force of gravity acts on the pendulum's mass in two ways. One way is tangentially, in which the force acts along the direction of the mass's motion. The other way is radially to the mass's direction of motion; in other words, along the pendulum's string. This can be shown with a diagram as follows.

As can be seen in the above diagram, the tangential acceleration is represented by the following expression.

Thus, as the pendulum swings to its lowest point, the value of 

Both the frequency and the period of the pendulum's harmonic motion is in no way related to the lowest point of the pendulum's path.

Finally, let's consider kinetic energy. We've already noted how the pendulum's potential energy is at a maximum at its highest point. As the pendulum falls to its lowest point, its potential energy is converted into kinetic energy. This is because as the pendulum falls to its lowest point, it speeds up more and more. Thus, at its lowest point, the pendulum has its kinetic energy at a maximum.

Page 2

Consider the following system:

If the length of the pendulum is 

Possible Answers:

Correct answer:

Explanation:

We can use the equation for conservation of energy to solve this problem.

If the initial state is when the mass is at its highest position and the final state is when the mass is at its lowest position, then we can eliminate initial kinetic energy and final potential energy:

Substituting expressions in for each term, we get:

Canceling out mass and rearranging for height, we get:

Thinking about a pendulum practically, we can write the height of the mass at any given point as a function of the length and angle of the pendulum:

Think about how this formula is written. The second term gives us how far down the mass is from the top point. Therefore, we need to subtract this from the length of the pendulum to get how high above the lowest point (the height) the mass currently is.

Substituting this into the previous equation, we get:

Rearrange to solve for the angle:

We have values for each variable, allowing us to solve:

Matt Damon is once again trapped on Mars. He must measure the length of rope he has using only a stopwatch. Please solve the problem below.

A pendulum on Mars has been measured to have a period of

Possible Answers:

Correct answer:

Explanation:

To find the answer one must manipulate the equation

Where

To solve this for

Now plugging in our numbers

Keep in mind that the most accurate method is to round numbers at the very end of calculations (above this isn't done from the start of pi for simplicity).

The unit calculation above will end with meters as we are taking 

Steven
Certified Tutor

University of Louisville, Bachelor of Science, Mechanical Engineering. University of Phoenix-Louisville Campus, Master of Eng...

Danielle
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Penn State University, Bachelor of Science, Civil Engineering.

Michael
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The University of Texas at Austin, Bachelors, Physics.

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