Chapter 6: Work and Kinetic Energy Suppose you want to find the final velocity of an object being acted on by a variable force. Newton s 2 nd law gives the differential equation (for 1D motion) dv dt = 1 m F (t) Often, however, the force is not known as a function of time, but as a function of position (springs, gravity, friction, etc). How to solve?
Work and Chapter Kinetic 6: Energy Work and Kinetic Energy Consider a force acting on a particle which moves along the s-axis. The force component F s causes the particle to speed up or slow down The work done on the particle due to this force is: The units of work are N m, where 1 N m = 1 kg m 2 /s 2 = 1 J. Slide 11-26
The Work-Kinetic Energy Theorem The net force is the vector sum of all the forces acting on a particle. The net work is the sum W net = ΣW i, where W i is the work done by each force. The net work done on a particle causes the particle s velocity to change. How does it change? Let s look at the situation for 1D motion: F net = ma = m dv Z dt W net = F net dx = Z m dv dt dx Slide 11-27
Work-Kinetic Energy Theorem Now think of the velocity as a function of position: dv dt = dv dx dx dt = dv Z dx v ) W net = m v dv Z dx dx = m vdv Integrating from the initial velocity to the final velocity, we get ) W net = 1 1 2 mv2 f 2 mv2 i Introducing kinetic energy, K = (1/2) m v 2, we get W net = K f K i
Conceptual Interpretation Work done by net force changes object s kinetic energy. If work is positive, kinetic energy increases If work is negative, kinetic energy decreases Example: Drop a rock off of a cliff of height h=15 m. What is the velocity right before it hits the ground? Net force due to gravity: F = -mg Force is constant, so net work is v f = W = 0 h K f = K i + mgh 1 2 mv2 f = mgh ( mg) dx = mgh 2gh = 17 m/s
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Relationship to Kinematics Equation Notice, that for a constant force, the work-kinetic energy theorem gives a familiar result: W = Fdx= K F x = 1 2 m(v2 f v 2 i ) v 2 f v 2 i =2a x Remember: this is only true for a constant acceleration (which requires a constant force).
Hook s Law The force a spring exerts on an object attached to one end is given (approximated) by Hook s Law: F s (x) = kx k is called the spring constant (determines how stiff the spring is) x is the displacement from the equilibrium position of the spring. Magnitude of force is the same on both ends of the spring.
Work Done by Spring Need to integrate force to determine work done by spring: W = xf x i F (x) dx = xf W = 1 2 k x2 i x i ( kx) dx Note: work is negative if final position is further from equilibrium position than initial position Book discusses work done in stretching or compressing a spring. Since there is no change in kinetic energy, the total work is zero and thus the work done by the external force is negative the work done by the spring. x 2 f
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Chalkboard Question (Example 6.7) An air-track glider of mass 0.1 kg is attached to the end of a horizontal air track by a spring with force constant 20.0 N/m. Initially the spring is unstretched and the glider is moving at 1.5 m/s to the right. If the coefficient of kinetic friction is 0.47, determine the maximum distance d that the glider moves to the right.
General Definition of Work (in 2D or 3D) So far we have been dealing with 1-dimensional situations. The general definition of work is given by W = P 1 P 2 F dl Dot product (chapter 1.10): A B = AB cos( )
Back to Chapter 1: Dot Products
Notice that, even for two or three dimensional problems, work is a scalar quantity! Work-Kinetic Energy Theorem still applies Example: Work done by a constant force acting at an angle to the displacement:
Work Done by Gravity on Ball on Incline What is the work done by gravity on a ball that rolls down a hill of incline of length L, as shown below? a) mgl b) mgh c) -mgl d) -mgh e) None of the above L
Work Done by Gravity on Ball on Incline Pick x-axis along ramp, with positive direction oriented down the ramp; F is constant and the trajectory is a straight line. Thus: if object moves a distance L, work done by gravity is W = mgl cos(ϕ) = mgl sin(θ) = mgh
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Clicker Question Three balls are thrown off a cliff with the same speed, but in different directions. Which ball has the greatest speed just before it hits the ground? A. Ball A B. Ball B C. Ball C D. A and B have same speed, but they are faster than C E. All balls have the same speed
Chalkboard Question A truck is driving down a freeway with an incline θ and slams on its breaks. Determine the stopping distance if the truck s initial velocity is v 1 and the coefficient of kinetic friction is µ k
Clicker Question I swing a ball around my head at constant speed in a circle with circumference 3 m. What is the work done on the ball by the 10 N tension force in the string during one revolution of the ball? A. 30 J B. 20 J C. 10 J D. 0 J
Note: Chalkboard Question YF 6.82: The rope and pulley have negligible mass, and the pulley is frictionless. The coefficient of kinetic friction between the 8-kg block and the tabletop is 0.250. The blocks are released from rest. What is the speed of the blocks after they have moved 1.5 m? Net work done on each object is equal to total change in kinetic energy of the system. What is net work done by the rope on the system?
Power Power is the rate of work being done. Power is not the same as force! P = dw dt Units: Joules/second = Watt; 1 hp = 746 W (= 550 ft lb/s) Luminosity is rate of energy being emitted in form of radiation. Same thing as power (e.g., 100 W light bulb) Mechanical power: W = F S P = W t = F S t = F v
Example: What is the maximum velocity of a 200 hp car with a cross-sectional area of 4 m 2? Air drag: F d = 1 4 Av2 For a car going at a constant velocity, the force exerted by the engine (via static friction) must equal (in magnitude) the the air drag force. F = F d = 1 4 Av2 Thus the power output of the car is P = 1 4 Av3 ) v = 4P A 1/3 = 53 m/s = 120 mi/h
Clicker question So the car in the previous question, with a maximum horsepower of 200 hp, has a maximum speed of 120 mi/h. What would the horsepower have to be if the maximum speed is to be 240 mi/h? a) 400 hp b) 800 hp c) 1200 hp d) 1600 hp
Chalkboard Question Suppose the same car (with a maximum horsepower of 200 hp) is now driving up a steep hill with a slope of 15. Ignoring air drag, what is the maximum speed the car can drive up the hill if its mass is 2,000 kg?
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Chalkboard Question Lance Armstrong used to exert a mechanical power of 400 Watts on average for an hour-long time trial. Assuming the body has an efficiency of 20% (only 20% of the energy spent by the body is used as mechanical energy), how much energy does Lance burn in one hour?
Clicker Question A person has a choice of hoisting up a crate of mass M using one of the two mechanical machines. He wishes to hoist the crate upwards a distance h. For which machine will the work done by the person be the smallest? A) The one on the le4 B) The one on the right C) The amount of work will be the same.
Chalkboard Question A person is hoisting up a crate of mass M using a mechanical machine. He is pulling down on the rope with a constant force such that the bottom of the rope is moving downward with a speed v. What is the mechanical power exerted by the person?