Mechanics and Heat. Chapter 5: Work and Energy. Dr. Rashid Hamdan

Transcription

1 Mechanics and Heat Chapter 5: Work and Energy Dr. Rashid Hamdan

2 5.1 Work Done by a Constant Force

3 Work Done by a Constant Force A force is said to do work if, when acting on a body, there is a displacement in the direction of the force. The work done by a constant force on an object moving along the x-axis: W F Work only involves the component of the force in the direction of the displacement. The SI units of work is Joules (J): 1J 1N.m 1kg.m /s Work is scalar and could be positive or negative depending on the relative direction of the force and displacement. A force in the same direction as the displacement does positive work. A force opposite direction to displacement does negative work. A force perpendicular to the direction of the displacement does zero work. x Δx

4 Work Done by a Constant Force T x f x n x is positive, T does a positive work on the sled. is negative, f does a negative work on the sled. and w are zero, n and w do no work on the sled. x The net work done on an object is the sum of the work done by individual force. Wnet W1 + W + W 3... W net F net, x Δx

5 General Rules: If a force makes an angle ϴ with the direction of motion then we can write: W (Fcosθ ) x If the net work done on the object is positive then its speed is increasing. If the net work is negative then its speed is decreasing. Zero net work means constant speed. Work done by friction or drag is always negative. If the motion is in dimensions then both components of the force can do work: W F Δx x + F Δy y

6 Work done by gravity: W g mg y The work done by gravity is positive if the object is going down and negative if the object is going up. Uniform Circular motion: In uniform circular motion the net force is always perpendicular to the direction of motion: The net work done on the object is zero. Speed is constant

7 5. Work Done by a Variable Force

8 Work from Force versus Position Graph The work done by a force with x-component F x is the area under the graph of F x versus position.

9 The Spring: Hook s law The displacement of the spring x is directly proportional to the force applied to the spring: F app, x kx The spring exerts a restoring force on body attached to it. F F kx (Newton s third Law) s, x app, x (Hook s Law) The proportionality constant (k) is called the spring constant and its SI units is N/m The spring constant is the slop of F x vs. x line

10 The Spring: Work Done The work done by the applied force on the spring is the area under the F app,x vs. x line. W applied 1 kx the work done by the applied force is positive. The restoring force exerted by the spring on the object attached to it is opposite to the displacement. Thus the work done by the restoring force of the spring is negative: W spring 1 kx

11 5.3 Kinetic Energy and Work Energy Theorem

12 Relating Net Work to Change in Speed: If a constant net force F net acts on an object doing a displacement Δx, the net work done on the object is : W F x net net, x Substituting the value of F from Newton s second law: Wnet ma xδx From the equation of motion: 1 a xδx (vx v0x ) Thus : 1 1 mv W mv Kinetic Energy K net x Kinetic energy is a scalar with SI units of joules. It is always positive and depend of the mass and speed of the object. K 1 mv K K ΔK (work-energy theorem) Wnet 0 0x (work-energy theorem)

13 Change in Kinetic Energy

14 5.4 Potential Energy

15 Conservative and Nonconservative Force W by spring ( 1 kx B ) ( 1 kx A ) The work done moving from A to B is the work done to go to B the work done to go to A The work done by kinetic friction depends on path taken.

16 Potential Energy The energy stored in a object due to its position relative to other objects is called Potential Energy. It s the ability of an object to do work due to a conservative force. Thus the work done is the negative of the change in potential energy: ΔU W conservative Gravitational Potential Energy ΔU mg y The gravitational potential energy Increase as the object moves upward And decrease the as the object moves down

17 Elastic Potential energy: As the spring is extended from position A to position B W by spring 1 kx A 1 kx B U -W by spring 1 kx B 1 kx A

18 The Zero of Potential Energy: The equations so far only defines the difference in potential energy between two positions. The assignment of a zero point of potential energy is truly arbitrary. Once a zero point is chosen, the potential energy at all other points is defined as the change from this point. For gravitational potential energy a good choice of the zero point is on the ground (y 0). Then: U g mgy For a spring it is best to assign U 0 at x 0, the springs equilibrium position: U 1 kx In different problems you might want to assign different zero point for potential energy. Physically only the change in potential energy is related to work and thus motion.

19 5.5 Conservation of Mechanical Energy

20 Conservation of Mechanical Energy: When there is no nonconservative forces acting on an object then: W ΔK net ΔU net The change in kinetic energy of an object is equal to negative the change in its potential energy or: ΔK + ΔU (K + U) 0 E K + U is the total mechanical energy and it is conserved when only conservative forces act on the object. E K + U constant This is known as the principle of conservation of mechanical energy.

21 Conservation of Mechanical Energy:

22 Nonconservative Forces Potential energy can only be defined for conservative forces. Force like friction and drag are nonconservative forces. Friction and drag forces do negative work reducing the kinetic energy of an object without increasing its potential energy. Thus reducing the total mechanical energy of the a system: E E + W final initial W f is the work done by friction or drag forces. Because W f is negative, E final is less than E initial f

23 Example A 65 kg downhill skier starts from rest at the top of a slope that drops 10 m vertically. At the bottom, she is moving at 3.5 m/s. Find the work done by frictional forces.

24 5.6 Power

25 Power The rate at which energy is expended or work is done is called Power (P). work energy delivered W P time time t The SI units for power is watt (W) 1W 1J/s Another common unit of power is the horsepower (hp) 1hp 745.7W Kilowatt-hour (KWh) is a common unit of energy. It is the energy consumed in 1 hour at a rate of 1 KW. 1KWh 3.6x10 6 J

26 Average and Instantaneous Power P work time Fx x t The average velocity is defined as The average power is given by: v x Fx x t P F x v x Average power The instantaneous power is defined by taking the limit as Δt goes to zero. P F x v x Instantaneous power

27 Example In these diagrams, the same car takes the time indicated to travel up the four hills shown. Each car drives at a constant speed, not necessarily the same speed as others. Neglecting friction rank in increasing order of the power required.

28 Chapter 5: Summary

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