Science One Physics Work and Energy

Transcription

1 Science One Physics Work and Energy Question 1: Why do we need a dot product? The energy of an isolated system is always conserved, but we can transfer energy to a system by interacting with it. One way to do this is by mechanical interactions, that is, by exerting a force. Let s try to understand how much we change the energy of an object by pushing on it. For simplicity, imagine we re talking about an isolated object in outer space, moving at some velocity! = (! $,! &,! ' ). The energy of this object is ) = +, -! $, + +, -! &, + +, -! ',. If we act with a force, this kinetic energy can change. We will now derive a relationship between rate at which K changes and the applied force. First, notice that the time derivative of K is a) Take the derivative of K with respect to time. which differentiation rules did we have to use to obtain this? b) show that we can rewrite this as: /0 = 2 /$ /1 $ + 2 /& /1 & + 2 /' /1 '. /1 By this equation, the rate of change of kinetic energy is related to the rate of change of position and the force. As a result, we can say that the change in energy during a short time interval will be related to the change in position by 3) = 2 $ & ' 36.

2 The expression on the right tells us how much energy we have added to an object by exerting a force on it. We call it the WORK done on the object by the force F 7 = 2 $ & ' 36. In general, the work can increase either kinetic energy or potential energy, or both. This equivalence, 7 89: = 3), is called the Work Kinetic Energy theorem, and holds true for the net force on an object, including those due to friction, tension, and potential energy. c) What would be a situation where exerting a force on something decreases its energy? Let s mention a few important properties of this expression for work: 1) We only do work (change the energy) if the object actually moves in the direction that the force is exerted in. If the object doesn t move, or if it moves perpendicular to the force (e.g. for a planet orbiting the Sun), there is no work done, so no change in energy. 2) The expression above is only valid if the force is constant over the path 3;. For a changing force, we need to break up the path into little segments and add up 2 $ & ' 36 for each segment. 3) The work is obtained by multiplying the common components of two vectors 2 = (2 $, 2 &, 2 ' ) and 3; = (34, 35, 36) and adding these up. This operation is known as the DOT PRODUCT 2 3; of two vectors, which results in a scalar. Geometrically, it is equal to the product of the lengths of the two vectors, times the angle between them. For two general vectors, we define: = > = = $ > $ + = & > & + = ' > '. And we can show that = > = = > cos B.

3 d) Show this by calculating = $ > $ + = & > & + = ' > ' for the vectors shown below in terms of the lengths A and B and the angle B. Question 2: Dot Product Practice Here are some happy vectors A = (2,1, 4), B = ( 3,0,1), and C = ( 1, 1,2). a) Calculate A B. b) Calculate the angle between A and B. c) State which of the following can be computed. I. A B C II. A (B C) III. A (B + C) IV. 3 B C Question 3: Will work for pizza

4 Since the beginning of the semester a friend of yours has been living with the worst roommate in the world. Your friend has finally managed to trade rooms and they ve asked you to help them move. In return for helping them move they re going buy you pizza. Your friend wants to make sure they buy enough pizza for you, so they want you to calculate how much energy you ll use moving the boxes. A slice of pizza has 200 Calories, and 1 Calorie is about 4200 Joules. a) The simplest way to move the boxes is to crouch down and push them horizontally along the floor. Each box weighs about 100 kg and the coefficient of friction between cardboard and dormitory floor linoleum is μ = 0.3. How much work do you do to push the box at constant velocity 30 m to the end of the hall? b) To make things easier, you find a rope to pull the box. You measure the tension in the rope with a spring scale, which reads 500 N. How much work do you do to move a 100 kg box 30 meters using the rope? c) Your friend now wants you to move a box up a short set of stairs. The figure below indicates two possible paths. Calculate the work done for each path.

5 d) Your friend now wants you to slide the box from one corner of the room to the other, as shown in the figure. Instead of doing it, you contemplate the two possible paths shown in the picture. Which path would you choose? Explain in terms of work and the dot product. e) Compare work calculated against friction forces (part d) and work calculated against gravity (part c). What do you notice about taking difference paths?

6 f) Your friend has 20 boxes to move. If you add up all the steps to move a box in parts b) and c), how much pizza should your friend buy so you can replenish your energy? Does this seem like enough pizza? Question 4: Potential Energy and Force An interesting result is that any conservative force (one in which the with is path independent) can be defined by a potential energy U(x,y,z). To find the component of the force in a given direction, we take the derivative of potential

7 energy with respect to that direction. For instance, the force in the x direction is given by 2 $ = /D /$. Let s see this at work. Below is a list of common forces. Identify each as either being conservative or non-conservative. If it s conservative, find the associated potential energy. If it s not, explain. a) The gravitational force 2 & = -E. b) The spring force described by Hooke s Law, 2 $ = F4. c) The friction force 2 $ = GH d) The tidal force. e) The tension force in a rope. f) For an electron in a hydrogen atom at distance r from the proton, the Coulomb force from the proton is 2 I = FJ, /;,, where k is a constant and e is the proton charge.

8 Question 5: Intermolecular Forces An intermolecular force is associated with the Leonard-Jones shaped potential below. Draw arrows at each of the dots below the x axis to show the direction and magnitude of the force on an object at the position of the dot. Try to be as accurate as possible.