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Chapter 7 Lecture 13: Kinetic Energy and Work: II HW5 (problems): 7.4, 7.15, 7.17, 7.31, 7.46, 7.63, 8.9, 8.31 Due Friday, March 10. Work Done by a Varying Force Assume that during a very small displacement, Dx, F is constant For that displacement, W ~ F Dx For all of the intervals, xf W Fx Dx xi Work Done by a Varying Force, cont lim Dx 0 xf F Dx x xi xf xi Fx dx xf Therefore,W Fx dx xi The work done is equal to the area under the curve between xi and xf Work Done By Multiple Forces If more than one force acts on a system and the system can be modeled as a particle, the total work done on the system is the work done by the net force W W net xf xi F dx x In the general case of a net force whose magnitude and direction may vary W W net xf xi Fdr Work Done by Multiple Forces, cont. If the system cannot be modeled as a particle, then the total work is equal to the algebraic sum of the work done by the individual forces Wnet Wby individual forces Remember work is a scalar, so this is the algebraic sum Work Done By A Spring A model of a common physical system for which the force varies with position The block is on a horizontal, frictionless surface Observe the motion of the block with various values of the spring constant Hooke’s Law The force exerted by the spring is Fs = - kx x is the position of the block with respect to the equilibrium position (x = 0) k is called the spring constant or force constant and measures the stiffness of the spring This is called Hooke’s Law Hooke’s Law, cont. When x is positive (spring is stretched), F is negative When x is 0 (at the equilibrium position), F is 0 When x is negative (spring is compressed), F is positive Hooke’s Law, final The force exerted by the spring is always directed opposite to the displacement from equilibrium The spring force is sometimes called the restoring force If the block is released it will oscillate back and forth between –xmax and xmax Work Done by a Spring Identify the block as the system Calculate the work as the block moves from xi = - xmax to xf = 0 xf 0 xi xmax Ws Fx dx kx dx 1 2 kxmax 2 The total work done as the block moves from –xmax to xmax is zero Work Done by a Spring, cont. Assume the block undergoes an arbitrary displacement from x = xi to x = xf The work done by the spring on the block is Ws xf xi 1 2 1 2 kx dx kxi kxf 2 2 If the motion ends where it begins, W = 0 Kinetic Energy Kinetic Energy is the energy of a particle due to its motion K = ½ mv2 K is the kinetic energy m is the mass of the particle v is the speed of the particle A change in kinetic energy is one possible result of doing work to transfer energy into a system Kinetic Energy, cont Calculating the work: W xf xi F dx xf xi ma dx vf W mv dv vi 1 2 1 2 W mv mv f i 2 2 Wnet K f K i DK 7.8: Work kinetic energy theorem with a variable force A particle of mass m is moving along an x axis and acted on by a net force F(x) that is directed along that axis. The work done on the particle by this force as the particle moves from position xi to position xf is : But, Therefore, Work-Kinetic Energy Theorem The Work-Kinetic Energy Theorem states SW = Kf – Ki = DK When work is done on a system and the only change in the system is in its speed, the work done by the net force equals the change in kinetic energy of the system. The speed of the system increases if the work done on it is positive The speed of the system decreases if the net work is negative Also valid for changes in rotational speed Work-Kinetic Energy Theorem – Example The normal and gravitational forces do no work since they are perpendicular to the direction of the displacement W = F Dx W = DK = ½ mvf2 - 0 Sample problem, industrial spies 7.6: Work done by gravitational force (a) An applied force lifts an object. The object’s displacement makes an angle f =180° with the gravitational force on the object. The applied force does positive work on the object. (b) An applied force lowers an object. The displacement of the object makes an angle with the gravitational force .The applied force does negative work on the object. 7.8: Work done by a general variable force B. Three dimensional force: If where Fx is the x-components of F and so on, and where dx is the x-component of the displacement vector dr and so on, then Finally, Instantaneous Power Power is the time rate of energy transfer The instantaneous power is defined as dE dt Using work as the energy transfer method, this can also be written as avg W Dt Power The time rate of energy transfer is called power The average power is given by W P Dt when the method of energy transfer is work Instantaneous Power and Average Power The instantaneous power is the limiting value of the average power as Dt approaches zero dW dr lim W Dt 0 F F v Dt dt dt The power is valid for any means of energy transfer Instantaneous Power and Average Power The SI unit of power is called the watt 1 watt = 1 joule / second = 1 kg . m2 / s3 A unit of power in the US Customary system is horsepower 1 hp = 746 W Units of power can also be used to express units of work or energy 1 kWh = (1000 W)(3600 s) = 3.6 x106 J Potential Energy Potential energy is energy related to the configuration of a system in which the components of the system interact by forces The forces are internal to the system Can be associated with only specific types of forces acting between members of a system 8.2 Work and potential energy The change DU in potential energy (gravitational, elastic, etc) is defined as being equal to the negative of the work done on the object by the conservative force (gravitational, elastic, etc) Gravitational Potential Energy The system is the Earth and the book Work is done on the book by lifting it slowly a vertical displacement D r Dyˆj The work done on the system must appear as an increase in the energy of the system Gravitational Potential Energy, cont There is no change in kinetic energy since the book starts and ends at rest Gravitational potential energy is the energy associated with an object at a given location above the surface of the Earth Gravitational Potential Energy, final The quantity mgy is identified as the gravitational potential energy, Ug Ug = mgy Units are joules (J) Is a scalar