Chapter 2.5 - The Dynamics Protocol

"The Protocol" (as we call it) for dynamics is a method for understanding and solving dynamics problems. You are not required to follow this protocol completely, but your protocol should include the basic steps in some sensible order.

 

Step 0: Identification

Before using any lens, you have to identify the most effective lens or lenses and provide motivation for it. Start by becoming familiar with the scenario you're exploring. Close your eyes and imagine it happening. Consider if you have any experience with anything like it. What was the outcome? Draw a good picture and indicate any relevant parameters. What's going on?

If the problem involves forces causing acceleration or change in momentum, then dynamics is often a good lens to look at the problem. State, "I'll use a dynamics lens because forces cause acceleration."

 

Step 1: Write LaTeX: \underline{\sum\vec{F}=m\vec{a}}F=ma_

Now identify the parts:

  • What is the object that is being accelerated by the forces?
  • What are the forces acting on this object (and what directions)?
  • What do you know about the acceleration of this object? Direction?

How do we best represent this? We make a Free Body Diagram (below). 

 

Step 2: Make a Free Body Diagram, and indicate acceleration

Free-body diagram for a block hanging over the edge of a counter.

Make a simple diagram of the chosen body showing all the forces acting on that body ONLY. That is, don't include forces the body exerts on something else. Make the forces start from the point they are acting from. For instance, in the FBD at right, we examine the small mass. We can draw the force of gravity coming from the center of mass, but the normal force is drawn from the place where the two blocks are in contact, where the normal force is being applied. Indicate the direction of acceleration, but do this off to one side, because the acceleration is NOT one of the forces. The acceleration is caused by the resultant force.

 

 

Step 3: Add the forces to get a resultant vector, the net force

Add the forces "nose to tail" like vectors and make a resultant vector from the very beginning to the very end. If you know the direction of the acceleration, this resultant vector must be in the same direction as the acceleration. If the body is in equilibrium, LaTeX: \vec{a}a = 0 and the point of the last vector should be at the very beginning, or "the snake bites its tail."

 

Step 4: Define the positive direction, and use  LaTeX: \underline{\sum\vec{F}=m\vec{a}}F=ma_

if you need to find a numerical answer.

 

 

Exercise 2.5.1

Please grade my solution for Example 2.4.1 in Chapter 2.4. How well did I follow the protocol? 

 

 

Gravity and Another Force: 

Bucket of water in space with normal force holding water in If we were floating in outer space, we would notice no 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: NN > 0. Given that this normal force is the only force, it would mean that the bucket would be accelerating in the same direction as the normal force. So, you could imagine accelerating the bucket back and forth or up and down while rotating it to maintain a positive normal force.  In Lacrosse, players make use of the motion as they cradle the ball in order to keep it in the pocket.

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

Noting that LaTeX: \sum\vec{F}=m\vec{a},F=ma,  consider the questions below as you keep a man, a ball, or water in a bucket as shown at right.

 

Exercise 2.5.2

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

  1. LaTeX: N>F_g.N>Fg.
  2. LaTeX: F_g>N>0.Fg>N>0.
  3. LaTeX: N>0.N>0.
  4. acceleration must be upward or zero only.
  5. acceleration can be zero or any value upward or downward.
  6. acceleration cannot be downward with a magnitude greater than gravity. 

 

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 chapter end. 

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

  1. LaTeX: N>F_g.N>Fg.
  2. LaTeX: F_g>N>0.Fg>N>0.
  3. LaTeX: N>0.N>0.
  4. acceleration must be downward with a magnitude greater than gravity.
  5. acceleration can be any value, but must be downward.
  6. acceleration can be zero or any value upward or downward.
  7. acceleration cannot be downward with a magnitude greater than gravity. 

 

 

Exercise 2.5.4

You are in an elevator that is accelerating upwards at 2 m/s2. Using your own mass, how much force is the elevator putting on you? Please follow the protocol: 

  • Identify Lens and motivation
  • Write down the operative equation
  • Draw FBD
  • Add the forces
  • Pick a positive direction and find the answer. 

 

Exercise 2.5.5

You are in an elevator and your mass is 50 kg. However, you find yourself standing upside down on the ceiling with the scale (between you and the ceiling) reading 100 N. What's the direction and magnitude of your acceleration? Please follow the protocol to convince me of your answer. 

 

 

Strings: The special thing about strings: if the applied force exceeds the breaking force, they snap and the tension immediately becomes zero. Because it only takes an instant to snap a string, the brief tension of the string on a body produces a tiny impulse ( LaTeX: \Delta\vec{p}=\vec{F}\Delta tΔp=FΔt), very little change in velocity, and a negligible displacement in the very small period of time. 

Assume that the strings in exercises 6-10 have a breaking tension of 300 N.

 

Exercise 2.5.6

Imagine a 10 kg iron sphere with a string tied to it in outer space. You pull on it with increasing force until it breaks.

  1. Make a graph of the force on the sphere as a function of time until after the string breaks.
  2. Make graphs of acceleration, velocity, and position of the sphere as a function of time until after the string breaks.
  3. Imagine if you are able to snap the string very quickly with an immediate tension > 300 N. Describe what you will see while watching the sphere. 

 

Exercise 2.5.7

Ball hanging from support by a string with a string hanging below Imagine a 10 kg iron sphere hanging on a string as shown at right. The same kind of string is also connected to the bottom of the sphere for you to pull on. Pull downward on the string with a force that increases very slowly until one string breaks. Which string breaks? What is the tension on the other string at this point? What is the acceleration of the ball before the string breaks? Immediately after the string breaks? In order to answer these questions: 

  • Assert that this is a dynamics lens, and support why this is the case.
  • Write LaTeX: \sum\vec{F}=m\vec{a},F=ma, then examine the forces and consider acceleration with a good FBD.
  • Show how the forces add to provide the net force on the ball before and after the string breaks. 

 

 

Exercise 2.5.8

Assume the same setup as in Exercise 2.5.7 above. However, this time, pull on the lower string with an immediate force greater than 300 N. Which string breaks? What is the tension on the other string at this point? What is the acceleration of the ball immediately before the string breaks? In order to answer these questions, follow either the protocol outlined in Exercise 2.5.7, or at the beginning of this chapter. 

 

 

Exercise 2.5.9

Assume the same setup as in Exercise 2.5.7 above. How could you break the top string without pulling on the bottom string at all? What is the acceleration of the ball before and after the string breaks? In order to answer these questions, follow either the protocol outlined in Exercise 2.5.7, or at the beginning of this chapter.

 

 

Exercise 2.5.10

Assume the same setup as in Exercise 2.5.7 above. However, this time, there is nothing pulling upward on the ball - only you pulling downward on a string attached to the ball. Can you still break the string pulling downward? If so, what would the acceleration of the ball be before and after the string breaks? In order to answer these questions, follow either the protocol outlined in Exercise 2.5.7, or at the beginning of this chapter.

 

 


Answer to Question 2.5.2: options III and VI are correct

Answer to Question 2.5.3 : options III and IV are correct