When you see something moving in a circle, it's a good bet that dynamics is involved. Make a free body diagram and note acceleration is towards the center of the circle. We will address circular motion when there is more than one force, such as in a roller coaster loop, where there is gravity and the normal force of the seat acting on your body!

A review from Chapter 2.5:

Bucket of water in space with normal force holding water in If we were in outer space, or in free fall inside the space station as it orbits the earth, there is no apparent force of gravity. How would we keep water inside a bucket, so that the water is touching the bottom of the bucket?

"Touching" the bottom of the bucket means that there is a nonzero normal force; that is LaTeX: N > 0. Given that this is the only force, it would mean that the bucket would be accelerating in the same direction. So, you could imagine accelerating the bucket back and forth or up and down while rotating it to maintain a positive normal force.

Please recall the elevator problem, noting that $LaTeX: \sum\vec{F}=m\vec{a}.$ Consider keeping a man, a ball, or water in a bucket as shown below.

Three buckets side-by-side. Left with man, middle with ball, right with water

Exercise 5.3.1 (same as Exercise 2.5.2)

If the object is kept in the bucket, what do we know? Identify all correct statements. Answers at end of Chapter 2.5.

acceleration must be upward or zero only.
acceleration can be zero or any value upward or downward.
acceleration cannot be downward with a magnitude greater than gravity.

Exercise 5.3.2 (same as Exercise 2.5.3)

How about if we see someone in a bucket upside down, what do we know? Identify all correct statements. Answers at end of Chapter 2.5.

Man in a bucket at the top of a vertical loop of the bucket

acceleration must be downward with a magnitude greater than gravity.
acceleration can be any value, but must be downward.
acceleration can be zero or any value upward or downward.
acceleration cannot be downward with a magnitude greater than gravity.

End of Review

Loop-the-loop with a flat ending When we see something moving in a circle, we only know that it is accelerating radially inward. We then look for the forces that could provide that acceleration according to $LaTeX: \sum\vec{F}=m\vec{a}.$ So, for instance with a "Loop-the-Loop" carnival ride, the forces include gravity and the normal force that cause centripetal acceleration inwards. So, at the top the centripetal acceleration is downward. If we want the cart to stay in contact with the track, we know that the normal force must be downward. What does this say about the centripetal acceleration resulting from gravity and the normal force?

Exercise 5.3.3

Please make a good free body diagram of the cart at the top of the loop and the bottom of the loop.
Please identify what you know for both the acceleration, and the normal force at the top of the loop and at the bottom of the loop.

Be sure to support your answer with a diagram and and lens identification.

Exercise 5.3.4

Diagram of a loop-the-loop with a ramp at the bottom of the loop, exercise 5.3.4. You go on a radius, LaTeX: R = 10 m, loop-the-loop ride where the cart is let go on a low friction track and is pulled downhill by gravity. You have to choose how high to start the cart. Say you have a mass of 70 kg and you are sitting on a scale that reads in kg. Don't use this drawing… please make your own.

If you start from a vertical height of 40 m, what does the scale under you read as you are at the top of the loop? What does it read at the bottom of the loop? Is this a good ride for pregnant women? How does it feel as you round the bottom of the loop?
What would happen if you decide to start the cart at the same height as the top of the loop? Why would this happen?
Please find the minimum vertical height, above the ground, that you must start the ride to stay on the track at the top.

The low friction, circular loop-the-loop ride above has a higher speed at the bottom than at the top. Thus, the centripetal acceleration to keep a circular path is also greater at the bottom. Additionally, please show that because of the way the track faces, in order to keep the cart on the track at the top, there must be a very great normal force at the bottom. This could make for a dangerous carnival ride! Is there some way we could increase the centripetal acceleration at the top (so the track stays connected to the cart), and lower the centripetal acceleration at the bottom (in order to not smoosh the people)?

Exercise 5.3.5

Photo of a carnival loop-the-loop ride, exercise 5.3.5. Look at the carnival ride at right. Notice that the radius of curvature at the top is not the same as the radius of curvature at the bottom. What effect does this have on the centripetal acceleration at the top and at the bottom? Why would they build it this way? Please support your answers with a good free-body diagram, and identify the lenses you use.

Exercise 5.3.6

Below, you see 3 pictures of a car you are driving. You are sitting on a common "bathroom scale" that is situated between your body and the seat.

Do you "feel" the same at the three different places? If not, how do you feel differently?
Does the scale read the same at all three locations? If so how do you know? If not, where is the reading the highest and where is the reading the lowest?
Please back up your answers above with a good free body diagram and indication of direction of acceleration.
In each of three places, how would the reading on the scale compare to the force of gravity on your body?

A red car at three different positions along a curved road, exercise 5.3.6.

Exercise 5.3.7

Do you weigh more at the North Pole?

You stand on a scale at the north pole and then on the same scale at the equator to see where the scale reads higher. Assume that the earth is a perfect rotating sphere. You find that at the equator, the scale reads:

More because the normal force compensates for both gravity and centripetal acceleration.
Less because the normal force at the equator is not enough to keep you in equilibrium.
The same because the normal force is always the same as gravity.
The same because you are in equilibrium in both places.
None of these.
Not enough information is given.

If you get confused… as with any mechanics problem, please draw a picture and do a lens analysis. There is a story that gold was mined in Alaska, weighed, and sent to Fort Knox near Washington DC where they weighed it again. What did they think in DC when they weighed the gold they’d just purchased?