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 

 term is at a minimum.

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 

 approaches zero. As it does this, the tangential acceleration also approaches zero.

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 

 and the maximum velocity of the block is , what is the minimum possible value of angle A?

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

seconds. Using the knowledge that gravity on Mars is  determine the length of the simple pendulum. Round to 3 significant figures.

Possible Answers:

 

Correct answer:

 

Explanation:

To find the answer one must manipulate the equation

Where

 represents the period of the motion,  the length of the pendulum, and  the gravity or acceleration the system is under.

To solve this for

 we will start by dividing both sides by . Next will with square both sides and finally multiply by , to come to the form below

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 

which will leave  as the final unit of your answer.

Steven
Certified Tutor

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

Danielle
Certified Tutor

Penn State University, Bachelor of Science, Civil Engineering.

Michael
Certified Tutor

The University of Texas at Austin, Bachelors, Physics.

If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources.

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC 101 S. Hanley Rd, Suite 300 St. Louis, MO 63105

Or fill out the form below: