Chapter 7 Kinetic Energy and Work

Forms of Energy Mechanical Mechanical focus for now focus for now chemical chemical electromagnetic electromagnetic nuclear nuclear

Using Energy Considerations Energy can be transformed from one form to another Energy can be transformed from one form to another Essential to the study of physics, chemistry, biology, geology, astronomy Essential to the study of physics, chemistry, biology, geology, astronomy Can be used in place of Newton’s laws to solve certain problems more simply Can be used in place of Newton’s laws to solve certain problems more simply

Work Done by a constant Force Provides a link between force and energy Provides a link between force and energy The work, W, done by a constant force on an object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement The work, W, done by a constant force on an object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement

Work, cont. F cos θ is the component of the force in the direction of the displacement F cos θ is the component of the force in the direction of the displacement Δ x is the displacement Δ x is the displacement

Work, cont. This gives no information about This gives no information about the time it took for the displacement to occur the time it took for the displacement to occur the velocity or acceleration of the object the velocity or acceleration of the object

Units of Work SI SI Newton meter = Joule Newton meter = Joule N m = J N m = J US Customary US Customary foot pound foot pound ft lb ft lb –no special name

More About Work Scalar quantity Scalar quantity The work done by a force is zero when the force is perpendicular to the displacement The work done by a force is zero when the force is perpendicular to the displacement cos 90° = 0 cos 90° = 0 If there are multiple forces acting on an object, the total work done is the algebraic sum of the amount of work done by each force If there are multiple forces acting on an object, the total work done is the algebraic sum of the amount of work done by each force

More About Work, cont. Work can be positive or negative Work can be positive or negative Positive if the force and the displacement are in the same direction Positive if the force and the displacement are in the same direction Negative if the force and the displacement are in the opposite direction Negative if the force and the displacement are in the opposite direction

When Work is Zero Displacement is horizontal Displacement is horizontal Force is vertical Force is vertical cos 90° = 0 cos 90° = 0

Work Can Be Positive or Negative Work done by F is positive when lifting the box Work done by F is positive when lifting the box

The Scalar Product of Two vectors Scalar product or dot product is a mathematical tool. Scalar product or dot product is a mathematical tool. A.B = |A|| B| cos(  ) A.B = |A|| B| cos(  ) |A| and |B| are the magnitudes of vectors A and B. |A| and |B| are the magnitudes of vectors A and B.  is the angle between A and B.  is the angle between A and B.

The Scalar Product of Two vectors Comparing A.B = |A||B| cos(  ), with Comparing A.B = |A||B| cos(  ), with We see that the work is the dot product between the force and the displacement: We see that the work is the dot product between the force and the displacement: W = F.  r

The Scalar Product of Two vectors The Scalar product is commutative: The Scalar product is commutative: A.B = B.A The scalar Product obey the distributive law of Multiplication: The scalar Product obey the distributive law of Multiplication: A.(B + C) = A.B + A.C A.(B + C) = A.B + A.C

The Scalar Product of Two vectors If A is perpendicular to B, then: If A is perpendicular to B, then: A.B = 0 It follows that the dot product between two units vectors is: Since the units vectors I, j and k are mutually perpendicular one to another. Also we have:

The Scalar Product of Two vectors Consider the following vectors:

Example: The vectors A and B are given by: The vectors A and B are given by: Determine the scalar product A.B Determine the scalar product A.B Find the angle  between A and B Find the angle  between A and B

Solution:  = 60.2°

Example: A particle moving in the xy-plane undergo a displacement in meters, as a constant force in Newtons acts on the particle. A particle moving in the xy-plane undergo a displacement in meters, as a constant force in Newtons acts on the particle. 1. Calculate the work done by F. 2. Calculate the magnitudes of F and  r. W = F.  r = 5   3 = 16 J W = F.  r = 5   3 = 16 J

Work Done by a varying Force Consider a particle being displaced along the x-axis under the action of a force that varies with position. Consider a particle being displaced along the x-axis under the action of a force that varies with position. In the small interval  x, F x can be considered to be constant and the work for the small displacement  x : In the small interval  x, F x can be considered to be constant and the work for the small displacement  x : W = F x  x W = F x  x

Work Done by a varying Force If we imagine that the F x versus x is divided into a large number of intervals, the total work done for the displacement from x i to x f is given by: If we imagine that the F x versus x is divided into a large number of intervals, the total work done for the displacement from x i to x f is given by:

Work Done by a varying Force When the number of small intervals is very large we can approximate the discrete sum (  ) into a continues integral: When the number of small intervals is very large we can approximate the discrete sum (  ) into a continues integral:

Work Done by a varying Force The work is the area under the curve. The work is the area under the curve.

Example: A force acting on a particle varies with x, as shown in the adjacent figure. Calculate the work done by the force as the particle moves from x = 0 to x=6.0 m. A force acting on a particle varies with x, as shown in the adjacent figure. Calculate the work done by the force as the particle moves from x = 0 to x=6.0 m.

Solution: The work is the area under the curve = Area of the trapezoid: The work is the area under the curve = Area of the trapezoid:

Example The adjacent Figure a) shows two horizontal forces that act on a block that is sliding to the right across a frictionless floor. Figure b) shows three plots of the block’s kinetic energy K versus time t. Which of the plots best corresponds to the following three situations: (a) F 1 = F 2, (b) F 1 > F 2, (c) F 1 < F 2 ?

Example: In the below figure, a greased pig has a choice of three frictionless slides along which to slide to the ground. Rank the slides according to how much work the gravitational force does on the pig during the descent, greatest first.

Work done by a spring: A block being pulled from x i = 0 to x f = x max on a frictionless surface by a force F app. The applied force is equal in magnitude and opposite in direction to the spring force F s at all times. A block being pulled from x i = 0 to x f = x max on a frictionless surface by a force F app. The applied force is equal in magnitude and opposite in direction to the spring force F s at all times. F s = -kx 

Work done by a spring: The work done by F app = -W s The work done by F app = -W s F app = kx 

Kinetic Energy Energy associated with the motion of an object Energy associated with the motion of an object Scalar quantity with the same units as work Scalar quantity with the same units as work Work is related to kinetic energy Work is related to kinetic energy

Work-Kinetic Energy Theorem When work is done by a net force on an object and the only change in the object is its speed, the work done is equal to the change in the object’s kinetic energy When work is done by a net force on an object and the only change in the object is its speed, the work done is equal to the change in the object’s kinetic energy Speed will increase if work is positive Speed will increase if work is positive Speed will decrease if work is negative Speed will decrease if work is negative

Work and Kinetic Energy An object’s kinetic energy can also be thought of as the amount of work the moving object could do in coming to rest An object’s kinetic energy can also be thought of as the amount of work the moving object could do in coming to rest The moving hammer has kinetic energy and can do work on the nail The moving hammer has kinetic energy and can do work on the nail

Example: A 6.0 kg block initially at rest is pulled to the right along a horizontal frictionless surface by a constant horizontal force of 12 N. Find the speed of the block after it has moved 3 m. A 6.0 kg block initially at rest is pulled to the right along a horizontal frictionless surface by a constant horizontal force of 12 N. Find the speed of the block after it has moved 3 m.

Solution: Neither of the vertically directed forces (mg and n) does work on the block, Since the displacement is horizontal. Applying the work kinetic energy theorem: Neither of the vertically directed forces (mg and n) does work on the block, Since the displacement is horizontal. Applying the work kinetic energy theorem: W F = F.  x =  KE  F.  x = ½ m(v f ) 2 – ½ m(v i ) 2 W F = F.  x =  KE  F.  x = ½ m(v f ) 2 – ½ m(v i ) 2 But v i = 0  x

Transferring Energy By Work By Work By applying a force By applying a force Produces a displacement of the system Produces a displacement of the system

Transferring Energy Heat Heat The process of transferring heat by collisions between molecules The process of transferring heat by collisions between molecules

Transferring Energy Mechanical Waves Mechanical Waves a disturbance propagates through a medium a disturbance propagates through a medium Examples include sound, water, seismic Examples include sound, water, seismic

Transferring Energy Electrical transmission Electrical transmission transfer by means of electrical current transfer by means of electrical current

Transferring Energy Electromagnetic radiation Electromagnetic radiation any form of electromagnetic waves any form of electromagnetic waves Light, microwaves, radio waves Light, microwaves, radio waves

Notes About Conservation of Energy We can neither create nor destroy energy We can neither create nor destroy energy Another way of saying energy is conserved Another way of saying energy is conserved

Power Often also interested in the rate at which the energy transfer takes place Often also interested in the rate at which the energy transfer takes place Power is defined as this rate of energy transfer Power is defined as this rate of energy transfer SI units are Watts (W) SI units are Watts (W)

Power, cont. US Customary units are generally hp US Customary units are generally hp need a conversion factor need a conversion factor The instantaneous power is defined as: