Lecture 11 Chapter 7 Physics I 10.16.2013 Work and Energy (Work Done by a Constant Force) Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html
Outline Chapter 7 Work done by a constant force Scalar (Dot) product of two vectors Kinetic Energy Work-Kinetic Energy theorem
Work Done by a Constant Force Work done on an object by a force If a constant force F moves object by a displacement d, work done by force is given by W = F d, where F is the component of the force along d d Work has units of N-m, or Joules (J), and is a scalar!!
Example I A force of 5 N, applied in the direction shown, moves a box across a horizontal distance of 10 m. How much work does the force do on the box? F 5N 60 d 10m = 25J
Scalar Product of Two Vectors Three ways to multiply vectors Multiplication by a scalar ca vector Scalar (or dot) product A B scalar Vector (or cross) product A B vector
Scalar (Dot) Product of Two Vectors The scalar product of two vectors is written as: Therefore, we can say that work is a scalar product of force and displacement Work done by a force along a displacement:
ConcepTest 1. To Work or Not to Work Is it possible to do work on an object that remains at rest? A) yes B) no Work requires that a force acts over a distance. If an object does not move at all, there is no displacement, and therefore no work done. The amount of work you actually do may have little relationship to the amount of effort you apply. For example, if you push on a car stuck in a snow drift, you may exert a lot of force (and effort) but if the car does not budge, you have not done any work! In order for work to be done on an object, the object must move some distance as a result of the force you apply.
Work F F d d F 90 d W 0 W 0 W 0
ConcepTest 2 A box is being pulled across a rough floor at a constant speed. What can you say about the work done by friction? Friction and Work I A) friction does no work at all B) friction does negative work C) friction does positive work Friction acts in the opposite direction to the displacement, so the work is negative. Or using the f N Displacement Pull definition of work (W = F d cos ), because = 180 º, then W < 0. mg
ConcepTest 3 Play Ball! In a baseball game, the catcher A) catcher has done positive work stops a 90-mph pitch. What can B) catcher has done negative work you say about the work done by C) catcher has done zero work the catcher on the ball? d W F 0 The force exerted by the catcher is opposite in direction to the displacement of the ball, so the work is negative. Or using the definition of work (W = F d cos ), because = 180 º, then W < 0. Note that because the work done on the ball is negative, its speed decreases. Follow-up: What about the work done by the ball on the catcher?
ConcepTest 4 A ball tied to a string is being whirled around in a circle. What can you say about the work done by tension? Tension and Work A) tension does no work at all B) tension does negative work C) tension does positive work No work is done because the force acts in a perpendicular direction to the displacement. Or using the definition of work (W = F d cos ), because = 180 º, then W < 0. T v Follow-up: Does the Earth do work on the Moon?
ConcepTest 5 A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box? Force and Work 1) one force 2) two forces 3) three forces 4) four forces 5) no forces are doing work Any force not perpendicular to the motion will do work: N does no work T does positive work f does negative work mg does negative work T N f mg
Work on a backpack. Example 7-2 (a) Determine the work a hiker must do on a 15.0-kg backpack to carry it up a hill of height h = 10.0 m, as shown. Determine also (b) the work done by gravity on the backpack, and (c) the net work done on the backpack. For simplicity, assume the motion is smooth and at constant velocity (i.e., acceleration is zero).
Work on a backpack. F H θ d 180 θ mg
Work on a backpack. F H d It is not an accident that the net work is zero. Wherever, the net force acting on an object is zero, the net work will be zero as well mg W net W1 W2 W3 F d F d F d F F F 2 3 1 2 3 d If F 0, then W 0 1 F net d net net
Scalar Product (component form) A k j A i A A z y x ˆ ˆ ˆ B k j B i B B z y x ˆ ˆ ˆ In component form: z z y y x x B A B A B A B A A B
Scalar Product Math Proof: ˆ ( i ˆ) j iˆ ˆj cos90 0 ˆ ( i iˆ) iˆ iˆ cos0 1 1 0 1 Using these we can write: AB (A x î A y ĵ A z ˆk)(Bx î B y ĵ B z ˆk) A x B x (î î ) A xb y (î ĵ) A xb z (î ˆk) A y B x ( ĵ î ) A yb y ( ĵ ĵ) A yb z ( ĵ ˆk) A z B x ( ˆk î ) A zb y ( ˆk ĵ) A zb z ( ˆk ˆk) ˆj Azkˆ ˆj B kˆ The scalar product of two unit vectors Similar (î ĵ) (î ˆk) ( ĵ ˆk) 0 and (î î ) ( ĵ ĵ) ( ˆk ˆk) 1 1 0 0 1 0 0 0 0 A Axiˆ Ay B Bxiˆ By A B A B A B 1 x x y y A z z B z
Properties of Scalar Product Commutative property A B B A C A B A C B A Distributive property
Example A constant force F acts on an object as it moves from position x 1 to x 2. What is the work done by this Force? x 1 d 2ˆ i F 4 ˆj x 4iˆ 5 ˆj 2kˆ x 3kˆ x x 2 1 2 5ˆ i 3 i ˆ 7 ˆ j 3 ˆj 9 k ˆ 6kˆ W F d ( 4)(3) (5)( 7) (2)(9) 29J
Translational Kinetic Energy Kinetic energy (K) is the energy of motion Translational kinetic energy is the energy of motion in a line or trajectory Rotational kinetic energy (Ch. 10) is the energy of circular/rotational motion Kinetic energy is the energy possessed by an object because of its motion
Kinetic Energy I A constant horizontal force F acts on an object on a frictionless surface. The box accelerates, i.e. speeds up. We want to define a property of the object (called kinetic energy, or energy of motion) which changes as work is done on it by the force. v i v f F d F
Kinetic Energy II v f 2 v i 2 2ad F ma 2 2 F v f vi 2 d m mv 2 f mv 2 i 2Fd 1 2 mv 2 f 1 2 mv 2 i Fd 1 2 mv 2 f 1 2 mv 2 i Fd K f K i W v i F Kinetic energy d v f K 1 2 mv2 Work-Kinetic Energy Principle K f K i K W net The work done is equal to the change in the kinetic energy. F
Work-Energy Principle Work-Kinetic Energy Principle K f K i K W net If the net work is positive, the kinetic energy increases. If the net work is negative, the kinetic energy decreases. Work and kinetic energy have the same units: joules. Energy can be considered as the ability to do work.
ConcepTest 6 Work and KE A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child? 1) positive work was done 2) negative work was done 3) zero work was done The kinetic energy of the child increased because her speed increased. This increase in KE was the result of positive work being done. Follow-up: What does it mean for negative work to be done on the child?
Example Net Work required to accelerate a 1000kg car from v i v f (a) from 20 m/s to 40 m/s? Work-Kinetic Energy Principle K f K i K W net W K m 2 [ v f v i 2 2 ] 1000 2 [40 20 2 ] 2 600,000J 600kJ (b) from 40 m/s to zero? m 2 2 1000 2 W K [ v f vi ] [0 40 ] 800,000J 800kJ 2 2
Work & Kinetic Energy The work done by the earth on the moon as the moon makes one full revolution around the earth at constant speed v is zero. So there is no change in kinetic energy. The work done by the earth on an apple as the apple falls towards the earth with increasing speed v is positive. So kinetic energy increases.
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