General Physics I. Lecture 4: Work and Kinetic Energy

Transcription

1 General Physics I Lecture 4: Work and Kinetic Energy

2 What Have We Learned? Motion of a particle in any dimensions. For constant acceleration, we derived a set of kinematic equations. We can generalized to the case a = a(t). Newton's Laws Applies to the constant gravitational force at the surface of the earth. Should also applies to gravitational force elsewhere in the universe, spring force, etc., which depend on the position of the particle. How do you solve the situation with a = a(x)?

3 The Most Generic Problem Generically, we can write For example, we have learned i Universal gravitation Drag force F i =F (x,v,t) F (v)= 1 2 F (r )=G Mm r 2 Dρ Av2

4 Solving Dynamics on a PC

5 A Little Improvement

6 Outline Introduce the concept of work done by a constant force. Develop the mathematical tool we need: the scalar product of two vectors Generalize the concept to work by a varying force. Introduce kinetic energy. Introduce the work-kinetic energy theorem. Discuss the concept of power: the time rate of doing work.

7 Work done by a Constant Force Definition: The work W done on an object by an agent exerting a constant force on the object is the product of the component of the force in the direction of the displacement and the magnitude of the displacement:

8 Different from Everyday Meaning A force does no work on an object if the object does not move. The work done by a force on a moving object is zero when the force applied is perpendicular to the object s displacement. Which force is doing non-zero work?

9 Work is a Scalar Quantity Work is a scalar quantity, whose SI unit is the N m, or the joule (J). Work is an energy transfer If energy is transferred to the system (object), W is positive. If energy is transferred from the system, W is negative. We need the scalar product of two vectors: force and displacement.

10 Scalar Product of Two Vectors In general, the scalar product (or the dot product) of any two vectors A and B is a scalar quantity equal to the product of the magnitudes of the two vectors and the cosine of the angle q between them:

11 Coordinate System

12 Exercise

13 Exercise

14 Properties of the Scalar Product Commutative law Distributive law of multiplication

15 Work Done by a Varying Force

16 Work Done by the Sun on a Probe Newton's law of universal gravitation will be discussed later.

17 Solution Does the path matter?

18 Hooke's Law Hooke s law: the force law for springs (in the limiting case of small displacements) x is the displacement of the block from its unstretched (x = 0) position. k is a positive constant called the force constant of the spring. The value of k is a measure of the stiffness of the spring.

19 Measure the Spring Constant

20 The Stretched Case The force exerted by the spring is always directed opposite the displacement.

21 The Negative Sign The negative sign means that the spring force always acts toward the equilibrium position (x = 0). It is, therefore, called a restoring force.

22 The Compressed Case The force exerted by the spring is always directed opposite the displacement.

23 Work Done by a Spring Suppose the block has been pushed to the left a distance x max from equilibrium and is then released.

24 Question Now let us consider the work done on the spring by an external agent that stretches the spring very slowly. What is the work done by this applied force (the external agent)? Hint: At any value of the displacement, the applied force is equal to and opposite the spring force.

25 Question What is the net work done by the spring force as the block moves from -x max to x max? Choices: (a) Positive (b) Zero (c) Negative

26 Question What is the net work done by the spring force as the block moves from -x max to x max? Choices: (a) Positive (b) Zero (c) Negative

27 Kinetic Energy Special case: constant acceleration Kinetic energy The net work done on a particle by a constant net force F acting on it equals the change in kinetic energy of the particle.

28 Work-Kinetic Energy Theorem The net work done on a particle by the net force acting on it is equal to the change in the kinetic energy of the particle. This is true whether or not the net force is constant. We can think of the kinetic energy as the work a particle can do in coming to rest, or the amount of energy stored in the particle.

29 Proof in the Generic Case Applying Newton's second law

30 What Matters?

31 VW EA288 4-Cylindar Diesel Engine 2.0 Liter, 112 kw (or rpm

32 Power Power: the time rate of doing work, or energy transfer. Average power: Instantaneous power: _

33 Unit of Power SI Unit of power: the watt (W) British engineering system: the horsepower (hp) Derived unit of energy: the kilowatt hour

34 Kinetic Energy at High Speeds Newtonian mechanics valid only for particle motion at small speeds (v << c). Therefore kinetic energy cannot approach infinity.