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Work and Energy Chapter 7 Conservation of Energy Energy is a quantity that can be converted from one form to another but cannot be created or destroyed. The total amount of energy present in any process remains the same. Physics chapter 7 2 Work Has a very specific definition in physics. Involves a force moving a body through a displacement. If the force is in the same direction as the motion, W Fs Physics chapter 7 3 Vector equation for work If the force is directed at an angle to the displacement, then only the component of the force that is parallel to the displacement does work. F W Fs cos Physics chapter 7 4 Vector equation for work Important – work is a scalar quantity, not a vector, even though it is derived from two vectors. Work does not have a direction. Physics chapter 7 5 Work If is between 0° and 90°, cos is positive, so work done is positive. If is between 90° and 180°, cos is negative, so work done is negative. If is 90°, cos is zero, so work done is zero. Physics chapter 7 6 Example Lifting, carrying, and lowering a book. Physics chapter 7 7 Units on work The unit for work is a joule. 1J 1 Nm Physics chapter 7 8 Example During your winter break you enter a dogsled race across a frozen lake. To get started you pull the sled (m = 80 kg) with a force of 180 N at 20° above the horizontal. You pull the sled 5 m. Find the amount of work you do Physics chapter 7 9 On your own A truck of mass 3000 kg is to be loaded onto a ship by a crane that exerts an upward force of 31 kN on the truck. This force is applied over a distance of 2 m. Find the work done by the crane on the truck Physics chapter 7 10 Work and energy with varying forces The equations we’ve developed so far for work involve constant forces and motion along a straight line. Now let’s look at varying forces. We could also look at motion along curved lines, but we would need to use integrals, which you don’t know how to do yet. Physics chapter 7 11 The spring force The farther you stretch a spring, the harder it is to stretch. The force you must apply follows the following equation, known as Hooke’s Law Fx kx Where k is called the force constant of the spring or the spring constant. k has units of N/m Physics chapter 7 12 Work done on a spring 1 2 W kX 2 Where X is the total elongation of the spring. See page 151. Physics chapter 7 13 Work done on a spring 1 2 1 2 W kx 2 kx1 2 2 Where x1 is the initial position of the spring, x2 is the final position of the spring x = 0 is the equilibrium position of the spring (when it is neither stretched nor compressed.) Physics chapter 7 14 Example A 4-kg block on a frictionless table is attached to a horizontal spring with a force constant of 400 N/m. The spring is initially compressed with the block at x1 = -5 cm. Find the work done on the block as the block moves to its equilibrium position x2 = 0 Physics chapter 7 15 On your own To stretch a spring 3.00 cm from its unstretched length, 12.0 J of work must be done. How much work must be done to compress this spring 4.00 cm from its unstretched length? Physics chapter 7 16 Work and Kinetic Energy The total work done on a body is related to changes in the speed of the body. v2 v 2as 2 2 1 v2 v1 a 2s 2 Physics chapter 7 2 17 Work and Kinetic Energy v2 v1 F ma m 2s 2 2 v2 v1 Fs m 2 1 1 2 2 Fs mv 2 mv1 2 2 2 Physics chapter 7 2 18 Work-Kinetic Energy Theorem 1 1 2 2 Wtot mv 2 mv1 2 2 The quantity ½ mv2 is called the kinetic energy and represented by the letter K. It is a scalar quantity and can never be negative. It is zero when an object is not moving. Wtot K2 K1 K Physics chapter 7 19 Units on kinetic energy Looking at the equation, we have 2 m kg 2 s m kg 2 m s N m Physics chapter 7 J 20 Work-kinetic energy theorem The work-kinetic energy theorem still works for varying forces. See the last paragraph on page 153 for a mathematical explanation of this. 1 1 2 2 Wtot mv 2 mv1 2 2 Physics chapter 7 21 Motion along a curved path W F dl P2 P1 Where dl is a distance along the path The work-kinetic energy theorem still holds true. Physics chapter 7 22 Example During your winter break you enter a dogsled race across a frozen lake. To get started you pull the sled (m = 80 kg) with a force of 180 N at 20° above the horizontal. You pull the sled for 5 m. Find The work you do The final speed of the sled Physics chapter 7 23 On your own A truck of mass 3000 kg is to be loaded onto a ship by a crane that exerts an upward force of 31 kN on the truck. This force is applied over a distance of 2 m. Find The work done by the crane on the truck The total work done on the truck The upward speed of the truck after the 2 m if it started from rest Physics chapter 7 24 Example A 4-kg block on a frictionless table is attached to a horizontal spring with a force constant of 400 N/m. The spring is initially compressed with the block at x1 = -5 cm. Find The work done on the block as the block moves to its equilibrium position x2 = 0 The speed of the block at x2=0 Physics chapter 7 25 Where does kinetic energy come from? If an object falls off a cliff, gravity does work on it as it falls, and increases its kinetic energy. But where does that energy come from? Physics chapter 7 26 Potential Energy Energy seems to be stored in some form related to height. This energy is related to the position of a body, not its motion. It is called potential energy. – measures potential or possibility for work to be done. (Some kinds of potential energy are related to things other than height.) Physics chapter 7 27 Work done by gravity Work done by gravity as a body falls from height y1 to height y2. Wgrav Fs cos0 Fs w y1 y2 mgy1 mgy2 Also works if object is rising – then y2 is greater than y1, so the work is negative, as it should be. Physics chapter 7 28 Gravitational Potential Energy We define the gravitational potential energy as U m gy So, the work done by gravity is Wgrav U1 U2 U 2 U1 Physics chapter 7 U 29 If only gravity does work Wtot Wgrav K U K 2 K1 U1 U 2 K 2 U 2 K1 U1 Physics chapter 7 30 Conservation of Mechanical Energy E K U constant When only gravity does work, the total mechanical energy is constant. Physics chapter 7 31 Where do I measure y from? Wherever you want. It is only the change in potential energy that we are interested in, not the value of U at a particular point. So choose your zero potential energy point wherever it is convenient. Physics chapter 7 32 Example Standing near the edge of the roof of a 12-m high building, you kick a ball with an initial speed of vi = 16 m/s at an angle of 60° above the horizontal. Neglecting air resistance, use conservation of energy to find How high above the height of the building the ball rises Its speed just before it hits the ground 9.79 m 22.2 m/s Physics chapter 7 33 You try You throw a 0.200-kg ball straight up in the air, giving it an initial upward velocity of 20.0 m/s. Use conservation of energy to find how high it goes. 20.4 m Physics chapter 7 34 Effect of other forces If forces other than gravity are acting on a body Wtot Wgrav Wother K 2 K1 U1 U 2 Wother K2 K1 K1 U1 Wother K2 U 2 Physics chapter 7 35 Example A child of mass 40 kg goes down an 8.0 m long slide inclined at 30° above the horizontal. The coefficient of kinetic friction between the slide and the child is 0.35. If the child starts from rest at the top of the slide, how fast is she traveling when she reaches the bottom? 5.60 m/s Physics chapter 7 36 Curved path The shape of the path makes no difference for gravitational potential potential energy. You can still use the same equations for conservation of energy. Physics chapter 7 37 On your own A pendulum consists of a bob of mass m attached to a string of length L. The bob is pulled aside so that the string make an angle 0 with the vertical, and is released from rest. Find an expression for its speed as it passes through the bottom of the arc. 0 Physics chapter 7 38 Elastic potential energy Think slingshot. The farther you stretch it, the more kinetic energy the projectile can gain. Physics chapter 7 39 Springs For springs, the elastic potential energy is given by 1 2 U kx 2 The work done by a spring is 1 2 1 2 Wel kx1 kx 2 U1 U 2 2 2 U The equations work whether the spring is stretched or compressed. Physics chapter 7 40 Where is x = 0? We don’t get to choose this time x = 0 is always at the equilibrium position of the spring – when it is neither stretched nor compressed. Physics chapter 7 41 Conservation of energy The conservation of energy works the same for springs as it does for gravity. K1 U1 Wother K2 U 2 Physics chapter 7 42 Conservation of energy If we have both gravitational and elastic potential energies, then K1 U grav ,1 Uel,1 Wother K2 U grav ,2 Uel,2 Physics chapter 7 43 Example A 1.20-kg piece of cheese is placed on a vertical spring of negligible mass and force constant k = 1800 N/m that is compressed 15.0 cm. When the spring is released, how high does the cheese rise from its initial position? (The spring and the cheese are not attached.) 1.72 m Physics chapter 7 44 On your own A 2000-kg elevator with broken cables is falling at 25 m/s when it first contacts a cushioning spring at the bottom of the shaft. The spring is supposed to stop the elevator, compressing 3.00 m as it does so. During the motion, a safety clamp applies a constant 17,000 N frictional force to the elevator. What is the force constant of the spring? 1.41 x 105 N/m Physics chapter 7 45 Conservative forces The work done by conservative forces: Can always be expressed as the difference between the initial and final values of a potential energy function. Is reversible Is path-independent (It only depends on the starting and ending points.) Equals zero when the starting and ending points are the same. Physics chapter 7 46 Conservative forces Gravitational (without air resistance) Spring (without friction) Physics chapter 7 47 Nonconservative forces Depend on path Are not reversible Cannot be expressed in terms of a potential energy function Do work even when the starting and ending points are the same Physics chapter 7 48 Nonconservative forces Friction Chemical reaction forces Physics chapter 7 49 Nonconservative forces Increase or decrease the internal energy of a system Frequently, this happens through heat. Uint Wother Physics chapter 7 50 Law of conservation of energy K U Uint 0 This is always true – the most general form of the equation. Physics chapter 7 51 Power Power is not the same as energy or force. Power is the time rate at which work is done. W Pav t Physics chapter 7 52 Power unit The unit of power is the watt J 1W 1 s Physics chapter 7 53 The kWh Your electric bill tells you how many kWh of electricity you used over the last month. This is a unit of energy or work, not a unit of power J 1000 3600s 6 s 1 kWh 3.6 10 J or 3.6 MJ 1 kW 1 h Physics chapter 7 54 Power in terms of velocity and force W Pav t Pav Fparallel s Fparallel t P Fparallel v Physics chapter 7 s Fparallel vav t P Fv cos 55 Example Force A does 5 J of work in 10 s. Force B does 3 J of work in 5 s. Which force delivers greater power? Physics chapter 7 56 On your own A 5-kg box is being lifted upward at a constant velocity of 2 m/s by a force equal to the weight of the box. What is the power input of the force? How much work is done by the force in 4 s? Physics chapter 7 57