a. Change of object s motion is related to both force and how long the force acts.

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1 0. Concept of Energy 1. Work. Power a. Energy is the most central concept underlying all sciences. Concept of energy is unknown to Isaac Newton. Its existence was still debated in the 1850s. Concept of energy is very difficult to define. b. Where is energy? Persons, places, and things have energy. We observe only the effects of energy when something is happening. When energy is being transferred from one place to another, or when energy is being transformed from one form to another. a. Change of object s motion is related to both force and how long the force acts. Impulse = Force x Time or J = Ft b. Definition: Work = Force x Distance or W = Fd c. The unit of measurement for work: newton-meter (N-m) = joule (J), 1J = 1N x 1m. (1 kg weights 9.8 N or kg m/s ). 1 KJ = 1,000 J, 1MJ = 1,000,000 J d. Two things happen where work is done: the application of a force and the movement of something by that force. If the work is done on an object, then the energy of that object changes. e. Work falls into two categories: work done against another force, and work done to change the speed of an object. f. Example 1: Calculate the work done when a 0-N force pushes a cart 3.5 m. W = Fd = 0 N x 3.5 m = 70 joules g. Example : Calculate the work done in lifting a 500-N barbell. m above the floor. What is the potential energy of the barbell when it is lifted to this height? W = Fd = 500 N x. m = 1,100 joules, PE = 1,100 joules. a. Definition: How fast the work is done or the rate at which the work is done. Work Done Power = Time Interval or P = W t = Fd t = F d t = Fv b. The unit of power is the joule per second, 1 watt (W) = 1 J / 1 second. 1 kilowatt (kw) = 1,000 watts, 1 megawatt (MW) = 1,000,000 watts. c. In metric system of units, automobiles are rated in kilowatts; in the United States, we rate engines in units of horsepower (hp). 1 hp = 0.75 kw; 134 hp = 100 kw. d. Example: Calculate the power expended when the barbell above is lifted. m in s. 3. Mechanical Energy 1,100 J / s = 550 watts Mr. Lin 1

2 a. Mechanical energy form: A compression of atoms in the material of an object, a physical separation of attracting bodies, or a rearrangement of electric charges in the molecules of a substance. b. This something that enables an object to do work is energy. Like work, energy is measured in joules. c. Two most common forms of mechanical energy: potential energy (PE) and kinetic energy (KE). Mechanical energy can be in the form of either one, or the sum of the two. (a) Potential energy the energy due to the position of something, or (b) Kinetic energy the energy due to the movement of something. 4. Potential Energy a. Definition: An object may store energy by virtue of its position. The energy that is held in readiness is called potential energy (PE), because in the stored state it has the potential for doing work. b. Examples of potential energy: a stretched/compressed spring, a drawn bow, a stretched rubber band, and the chemical energy in fuels (at submicroscopic level). Any substance that can do work through chemical action possesses potential energy (such as fossil fuels, electric batteries and food). c. Work is required to elevate objects against Earth s gravity. The potential energy due to elevated positions is called gravitational potential energy (GPE). The amount of gravitational potential energy possessed by an elevated object is equal to the work done against gravity in lifting it. The upward force required while moving at constant velocity is equal to the weight (mg) of the object. Gravitational Potential Energy = Weight x Height or GPE = F g h = mgh or PE = mgh d. The gravitation potential energy depends only on mg and h, does not depend on the path. e. Hooke s Law: Hooke's Law says that the spring restoration force (F S ) due to a spring is proportional to the length that the spring is stretched or compressed, and acts in the opposite direction. F S = kx k is spring constant, and x change in spring length from the equilibrium position f. Spring Potential Energy: A spring may store energy due to compression or stretching and is called spring potential energy (PE S ). The potential energy stored in spring is equal to the work done to the spring: PE S = F S x = ( 1 F S )x = ( 1 kx)x = 1 kx or PE S = 1 kx k x is spring constant, and change in spring length from the equilibrium position 5. Kinetic Energy a. If an object is moving, then it is capable of doing work. It has energy of motion, or kinetic energy (KE). The kinetic energy of an object depends on the mass and speed of the object. Kinetic Energy = 1 mass speed or KE = 1 mv b. The kinetic energy of a moving object is equal to the work required to bring it to that speed from the rest, or the work the object can do while being brought to rest. Mr. Lin

3 Net Force x Distance = Kinetic Energy or KE = Fd = (ma)( 1 at ) = 1 m(at) = 1 mv c. If the speed of an object is doubled (the speed is squared), its kinetic energy is quadrupled. It takes four times the work to double the speed. It takes four times of work to stop a double-speed object. d. Work-Energy Theorem: Work changes energy. If no change in energy occurs, then no work is done. Work = ΔE 6. Conservation of Energy a. Energy changes from one form to another. It transformed without net loss or net gain. b. Law of conservation of energy: Energy cannot be created or destroyed. It can be transformed from one form into another, but the total amount of energy never changes. c. Energy transform in a pendulum: The PE of the pendulum bob at its highest point is equal to the KE of the bob at its lowest point. Everywhere along its path, the sum of PE and KE is the same. 7. Case Study: Conservation of Energy a. A mass, m, is hung from an ideal spring (its length is equal to zero and no friction) and reaches equilibrium at position B when the spring has been stretched to the length l. The mass is then pulled to position C and released. The position C is of m l height h below the position B. The mass oscillates between positions A and C. d [Neglect friction.] (a) At which position, A, B, or C, is mass m located when the kinetic energy, gravitational potential energy, or elastic potential energy of the m system is at a maximum? (b) Prove that position A is of height h above the position h B. (c) Calculate the speed (v B ) of the mass m at position B when released from the m position C. (a) Position B maximum kinetic energy, since the mass has been pushed form position A or pulled from position C by the spring all the way until position B. The mass has continued accelerating until position B, and then started experiencing the opposite spring force and started de-accelerating. So, at position B, the mass has largest speed, and thus, has the greatest kinetic energy (KE = ½ mv ). Position A maximum gravitational energy, since the height of position A is the largest, so, it has the largest gravitational energy (GPE = mgh). Position C maximum spring potential energy, since the spring has been stretched the longest at position C, so, the spring potential energy is the largest (PE S = ½ kx ). (b) When the mass is in equilibrium, the spring force is equal to the weight: F g = F S, or mg =kl. So, the spring constant k = mg/l. Due to conservation of energy, the total energy of the mass at position A, B and C will be all the same: A B C PE A + PE SA + KE A = PE B + PE SB + KE B = PE C + PE SC + KE C Assume the position A is d higher than the position B, and the gravitational potential energy at the position C is zero, then, mg(h+d) + ½ k(l-d) + 0 = mgh + ½ kl + ½ m v B = 0 + ½ k(l+h) + 0 Now, we can reorganize the equations: mg(h + d)+ 1 k(l = 1 k(l + d) h) mg(h + d)+ 1 k(l = mgh + 1 d) kl + 1 mv B By substituting k by mg/l, and expanding and simplifying the equations: Mr. Lin 3

4 mg(h + d)+ mg l (l = mg (l + d) h) l mg(h + d)+ mg l (l = mgh + mg d) l l + 1 mv B d = h g l d = v B d = h v B = h g l So, we prove that d = h. (c) We also find v B : v B = h g l b. Base on the above context and discussions, if the stretch distance of the spring is x, (a) represent the total mechanical energy (TE), gravitational potential energy (PE), spring potential energy (PE S ), kinetic energy (KE) and velocity (v) in terms of the mass (m), acceleration due to gravity (g), spring equilibrium length (l), spring pull-down distance (h), and spring stretch distance (x). (b) Graph the PE, PE S, KE and TE vs. x and explain the relations among them. (a) The total energy (TE) is the same everywhere due to the law of conservation of energy. So, TE = 1 k(l + = mg (l + h) h) l PE = mg (l + h x) PE S = 1 kx = mg l x KE = 1 mv = TE PE PE S So, solve for v: v = g l [h (l x) ] when x = l, v has maximum value: = mg l (l + h) mg(l + h x) mg l x = mg l [h (l x) ] v = h g l (b) The total energy (TE) is a constant (horizontal line) and is independent of the stretch distance (x). TE = mg (l + h) l The gravitational potential energy (PE) is a linear function of stretch distance (x) with negative slope. When PE = 0, the x-intercept is (l+h); when x = 0, the y-intercept is mg (l+h). PE = (mg)x + mg (l + h) The spring potential energy (PE S ) is a quadratic function of upward-opening parabola. When x = (l+h), PE S = TE and PE S intersects with TE. When x = 0, PE S = 0. PE S = mg l x The kinetic energy (KE) is a quadratic function of downward-opening parabola. When x = (l+h) and x = (l-h), KE = 0. KE = ( mg l )x +(mg)x + mg l (h l ) And when x = l (axis of symmetry), KE has the maximum value: KE = mg l h Energy (E) mg(l+h) mg(l+h) /(l) mgh /(l) 0 A B C l h PE KE l PE S l+h TE Stretch Dist. x Mr. Lin 4

5 8. Machines a. A device used to multiply forces or simply to change the direction of forces is called a machine. The concept that underlies every machine is the conservation of energy. The work input will be equal to the work output. Work Input = Work Output or (Force x Distance) input = (Force x Distance) output b. A lever can multiply forces, but no machines can multiply work or energy. c. Mechanical advantage: the ratio of output force to input force for a machine. 9. Lever Machine Advantage = Output Force Input Force = Input Distan ce Output Distan ce Three types of lever: Type Type 1 Type Type 3 Between force and load Load is between the The fulcrum is at one end Fulcrum Position (input and output) fulcrum and the input force and the load is at the other Diagram Example Direction of input and output Usage Playground seesaw, Balance, Scissor Metal bar to lift car, Nutcracker, Bottle Opener, Wheelbarrow Opposite Same Same Increase force at the expense of distance Increase force at the expense of distance Biceps muscles, Tweezers, Fishing Rod Increase distance at the expense of force 10. Pulley a. A pulley is a kind of lever that can be used to change the direction of a force and multiply forces. b. Two types of pulley: Type Fixed Moving Diagram Similarity to Lever Type 1 lever Type lever Input Output Direction Opposite Same Usage Change direction of a force Multiply force Mechanical Advantage 1 I/P Dist. : O/P dist. 1:1 :1 c. The mechanical advantage of a pulley system is the number of strands of rope which support the load. d. The work input to the pulley system should be equal to the work output, and work is equal to force times distance. MA = 1 MA = MA = 3 Mr. Lin 5 MA = 4

6 Work input = Work output or Force input x Distance input = Force output x Distance output 11. Efficiency a. Ideal machine transfers all the work input to work output, the efficiency would be 100%. In practice, 100% efficiency never happens. Some energy is transformed into atomic or molecular kinetic energy (thermal energy heat is energy being transferred from one place to another by atomic and molecular motion). The lower the efficiency of a machine, the greater is the amount of energy wasted as heat. Efficiency is always less than one. b. Definition: Efficiency = Useful work output Total work input c. The mechanical advantage (MA) of a machine is: MA = Force output Force input = Distance input Distance output d. The ideal (or theoretical) mechanical advantage (IMA or TMA) of a system is only related to the geometry and configuration of the machine: IMA (or TMA) = Distance input Distance output e. In real-life, some of the work input are consumed by the system frictions and the other factors. Only partial of the work are useful to do the job. So, the actual mechanical advantage (AMA) is: AMA = Force output Force input f. The efficiency of a machine: work input are consumed by the system frictions and the other factors. Only partial of the work are useful to do the job. So, the efficiency is: Efficiency = Work useful Work input = Force output Distance output Force input Distance input = AMA IMA g. Only 30% of the energy produced by burning gasoline in a typical automobile engine becomes useful mechanical energy. 1. Inclined Plane a. An inclined plane is a machine and sliding a load up an incline requires less force than lifting it vertically. b. Mechanical advantage of inclined plane = Length of inclined plane Height of inclined plane c. Examples: Auto Jack and crew. Jack is an inclined plane wrapped around a cylinder. If circular distance the handle is moved is n times greater than the pitch, than the theoretical mechanical advantage of the jack is n. If the efficiency is e, the actual mechanical advantage = n x e. 13. Energy for Life a. The thermonuclear fusion causes hydrogen nuclei to fuse and become helium nuclei. The process release radiant energy, some of which reaches earth. Mr. Lin 6

7 b. Most living organisms on this planet feed on various hydrocarbon compounds that release energy when they react with oxygen. c. There is more energy stored in the molecules in food than there is in the reaction products after the food is metabolized. The energy difference is what sustains life. d. During metabolism, the reaction rate is slower and energy is released as it is needed by the body. The reaction is self-sustaining once it starts. e. The reverse process is photosynthesis and requires energy input (sunlight). 14. Work and Energy Problems: a. A rollercoaster cart leaves point A of known height h A m at a given speed of v A m/s, passes through point B on the ground, and reaches point C at a speed of v C m/s. The total mass of the cart assembly and the passengers are m kg, and there is no friction in the system. (a) What is the speed (v B ) of the cart at point B? (b) What is the height (h C ) of point C? v A m/s A h A m B v B m/s C h C m v C m/s b. If you throw a ball with mass m kg straight upward at a speed of v m/s from the top of a building, and the building is h m tall. When the ball returns to you, you should not catch it and let the ball continue falling to the ground without hitting any thing. (a) How much works have you done to the ball? (b) What s the maximum height will the ball reach? (c) What s the gravitational potential energy of the ball when it reaches the highest point? (d) What s the speed of the ball when it falls to the half-height of the building? (e) What s the final speed of the ball right before the ball hit the ground? h m Mr. Lin 7

8 c. A pendulum is released from rest at point A of height h m. When it reaches point B of height h m, the speed is v B m/s. (a) What is the speed (v C m/s) of the pendulum when it reaches the lowest position of point C? (b) What is the speed (v D m/s) of the pendulum at point D of height ½ h m? A h m B h m v B m/s C D v D m/s ½ h v C m/s d. A block with mass m kg is moving at constant velocity v m/s on a frictionless surface. If the block hit an ideal spring and compress the spring for x m right before the block bounce back. If there is no energy loss during the process, what is the spring constant k? v m/s m kg x m e. A F-N force is applied to an m-kg cart at rest for t seconds on a frictionless surface. (a) What s the momentum change of the cart after t seconds? (b) How much work has been done to the cart? (c) What s the average power in the first t seconds? (d) What s the kinetic energy of the cart after t seconds? F N m kg Mr. Lin 8