WORK

1. WORK And Energy and Power

2. What is work? • A body experiences a change in energy when one or more forces do work on it. • A body must move under the influence of a force or forces to say work was done. • A force does positive work on a body when the force and the displacement are at least partially aligned. (Maximum positive work is done when a force and a displacement are in exactly the same direction). • If a force causes no displacement, it doeszero work. • Forces can do negative workif they are pointed opposite the direction of the displacement.

3. Forces can do positive or negative work: • When the load goes up, gravity does negative work and the crane does positive work. • When the load goes down, gravity does positive work and the crane does negative work. F mg

4. Calculating Work: If a force on an object is at least partially aligned with the displacement of the object, positive work is done by the force. The amount of work done depends on: The magnitude of the force The magnitude of the displacement The degree of alignment W= Frcosθ cos(0) = ? cos(90) = ? F F   r

5. Scalar Dot Product? A product is obviously a result of multiplying 2 numbers. A scalar is a quantity with NO DIRECTION. A dot product is basically a CONSTRAINT on the formula. In this case it means that F and x MUST be parallel. To ensure that they are parallel we add the cosine on the end. So basically Work is found by multiplying the Force times the displacement and the result is ENERGY, which has no direction associated with it. Units of Work: • SI System: • Joule (Nm)

6. WORK

7. F m Work and Simple Machines • A pulley system, which has at least one pulley attached to the load, can be used to reduce the force necessary to lift a load. • Amount of work done in lifting the load is not changed. • The distance the force is applied over is increased, thus the force is reduced, since W = Fd. • What about incline planes (or levers?)

8. Units of Work: • SI System: • Joule (Nm) • British System: • foot-pound • cgs System: • erg (dyne-cm) • Atomic Level: • electron-Volt (eV)

9. Practice Problem: A block of mass 2.50 kg is pushed 2.20 m along a frictionless surface with a constant 16.0 N force directed at an angle of 25.0° below the horizontal. Determine the work done by: • The Applied Force • The Normal Force • The Force due to gravity • Find the net force done on the block 31.9 J 0 J 0 J 31.9 J

10. Practice Problem: A block of mass 5.0 kg starts from rest and slides 2.50 m down an inclined plane. The coefficient of kinetic friction between the block and the plane is 0.436. Determine the work done by: • Gravity • Friction • The Normal Force 61.3 J -46.3 J 0 J

11. Practice Problem: Vector A has a magnitude of 8.0 and vector B has a magnitude of 12.0. The two vectors make an angle of 40o with each other. Find A•B. 73.5 J

12. Practice Problem: A force F = (5.0i + 3.0j) N acts upon a body which undergoes a displacement d = (2.0i – j) m. How much work is performed, and what is the angle between the vectors? W = 7 J at 56.7°

13. Practice Problem: A force F = (5.0i + 6.0j – 2.0k)N acts on an object that undergoes a displacement of r = (4.0i – 9.0j + 3.0k)m. How much work was done on the object by the force? (Find the magnitude and direction) W = -40 J at 118.6°

14. Work and Variable Forces • For constant forces • W = F • r • For variable forces, the force is only constant over an infinitesimal displacement • dW = F • dr • To calculate work for a larger displacement, you have to take an integral • W = dW = F • dr

15. F(x) x xa xb Work and Variable Forces The area under the curve of a graph of force vs displacement gives the work done by the force.

16. Work and Variable Forces • The function here must be a function with respect to “x” or “r”. • Let’s look at a POPULAR force function (F=ma). • Is this function with respect to “x”? NO! • You can still integrate the function, it simply needs to be modified so that it fits the model accordingly.

17. Practice Problem: A force A force acting on a particle varies with x, as shown. Calculate the work done by the force as the particle moves from x = 0 to x = 6.0 m. W = 25 J

18. Practice Problem: A force acting on a particle is Fx= (4x – x2)N. Find the work done by the force on the particle when the particle moves along the x-axis from x= 0 to x = 2.0 m. W = 5.3 J

19. Work Done By a Spring The Force Required to stretch or compress a spring is: F = -kx Where k is the spring constant and x is the amount of stretch or compression. This is known as Hooke’s Law

20. Work Done By a Spring The negative sign in F=-kx means that the force exerted by the spring is always opposite of the displacement. If we compress a spring from some arbitrary position xi to some position xf then the work done on the object is (This is also the potential energy stored in the spring):

21. Practice Problem: A spring is hung vertically, and an object of mass 0.55 kg is attached to its lower end. Under the action of the “load” mg, the spring stretches a distance of 2.0 cm from its equilibrium position. • Find the spring constant of the spring • Find the Work done by the spring as it stretches through this distance k = 2.7 x102 N/m W = -5.4 x 10-2 J

22. ENERGY A change in energy is WORK If you apply a FORCE on an object and it covers a displacement you have supplied ENERGY or done WORK on that object.

23. ENERGY: There are many different TYPES of Energy • Energy is expressed in JOULES (J) • 4.19 J = 1 calorie • Energy can be expressed more specifically by using the term WORK(W)

24. Kinetic Energy • Kinetic energy is one form of mechanical energy. Kinetic energy is due to the motion of an object. • K = ½ m v2 • K: Kinetic Energy in Joules. • m: mass in kg • v: speed in m/s • In vector form, K = ½ mv•v

25. Net Work or Total Work • An object can be subject to many forces at the same time - one force may be doing positive work on the object, another force may be doing negative work, and yet another force may be doing no work at all. • The net work, or total, work done on the object (Wnet or Wtot) is the scalar sum of the work done on an object by all forces acting upon the object. • Wnet = ΣW

26. The Work-Kinetic Energy Theorem • Wnet = ΔK • When net work due to all forces acting upon an object is positive, the kinetic energy of the object will increase. • When net work due to all forces acting upon an object is negative, the kinetic energy of the object will decrease. • When there is no net work acting upon an object, the kinetic energy of the object will be unchanged. • (Note this says nothing about the kinetic energy.)

27. Work-Kinetic Energy Theorem An important relationship between kinetic energy and work can be shown with the derivation below The net work done on a particle equals the change in its kinetic energy

28. Practice Problem: A 70 kg base-runner begins to slide into second base when moving at a speed of 4.0 m/s. The coefficient of kinetic friction between his clothes and the earth is 0.70. He slides so that his speed is zero just as he reaches the base • How much energy is lost due to friction acting on the runner? • How far does he slide? -560 J 1.17 m

29. Something interesting…. Consider a mass m that moves from position 1 ( y1) to position 2 m,(y2), moving with a constant velocity. How much work does gravity do on the body as it executes the motion? Suppose the mass was thrown UPWARD. How much work does gravity do on the body as it executes the motion? In both cases, the negative sign is supplied

30. Conservative Forces vs Non-Conservative Forces The amount of Work gravity does on a body is PATH INDEPENDANT. Force fields that act this way are CONSERVATIVE FORCES FIELDS. The total amount of work done on a body that moves around a CLOSED PATH in the field will always be ZERO with conservative forces FRICTION is a non conservative force. By NON-CONSERVATIVE we mean it DEPENDS on the PATH. If a body slides up, and then back down an incline the total work done by friction is NOT ZERO. When the direction of motion reverses, so does the force and friction will do NEGATIVE WORK in BOTH directions.

31. Power • Power is the rate of which work is done. • One of the things we do everyday is measure how much energy we use. The rate at which we use it determines the amount we pay to our utility company. Since WORK is energy the rate at which work is done is referred to as POWER. • No matter how fast we get up the stairs, our work is the same. • When we run upstairs, power demands on our body are high. • When we walk upstairs, power demands on our body are lower.

32. Power • Pave = W / t • Pinst = dW/dt • P = F • v UNITS OF POWER • Watt = J/s • ft lb / s • horsepower • 550 ft lb / s • 746 Watts