5 Gravitational and Elastic Potential Energy PDF

UNIT V Gravitational Potential Energy, Elastic Potential Energy(7.1 to 7.5) GRAVITATIONAL POTENTIAL ENERGY Diver jumping off a high board into a swimming pool There is gravitational potential energy even when the diver is at rest on the high board....

UNIT V Gravitational Potential Energy, Elastic Potential Energy(7.1 to 7.5) GRAVITATIONAL POTENTIAL ENERGY Diver jumping off a high board into a swimming pool There is gravitational potential energy even when the diver is at rest on the high board. As she falls, this potential energy is transformed into her kinetic energy. If the diver bounces on the end of the board before she jumps, the bent board stores a second kind of potential energy called elastic potential energy.  GRAVITATIONAL POTENTIAL ENERGY Energy associated with position is called potential energy. The potential energy associated with a body’s weight and its height above the ground is called gravitational potential energy When a body is raised to a height , it stores energy and have the potential for the gravitational force to do work on it, but only if it falls down to the ground Thus a falling body’s kinetic energy increases because the force of the earth’s gravity does work on the body. The other way is to say that the kinetic energy increases as the gravitational potential energy decreases  GRAVITATIONAL POTENTIAL ENERGY Suppose a body with mass m moves along the (vertical) y-axis. The forces acting on it are its weight, with magnitude w = mg, and possibly some other forces; we call the vector sum (resultant) of all the other forces 𝐹𝑜𝑡ℎ𝑒𝑟. The work done by the weight when the body moves downward from a height y1 above the origin to a lower height y2. The weight and displacement are in the same direction, so the work 𝑊𝑔𝑟𝑎𝑣 done on the body by its weight is positive: When the body moves upward and y2 is greater than y1, the quantity y1 - y2 is negative, and Wgrav is negative because the weight and displacement are opposite in direction. GRAVITATIONAL POTENTIAL ENERGY GRAVITATIONAL POTENTIAL ENERGY  CONSERVATION OF MECHANICAL ENERGY WHEN GRAVITATIONAL FORCE ALONE ACTS Suppose a body’s weight is the only force acting on it, so 𝐹𝑜𝑡ℎ𝑒𝑟 = 0. The body is then falling freely with no air resistance and can be moving either up or down. Let its speed at point y1 be v1 and let its speed at y2 be v2. The work–energy theorem says that the total work done on the body equals the change in the body’s kinetic energy: 𝑊𝑡𝑜𝑡 = ∆𝐾 = 𝐾2 − 𝐾1. If gravity is the only force that acts, then 𝑊𝑡𝑜𝑡 = Putting these together, we get The sum K + Ugrav of kinetic and potential energies is called E, the total mechanical energy of the system  CONSERVATION OF MECHANICAL ENERGY But since positions y1 and y2 are arbitrary points in the motion of the body, the total mechanical energy E has the same value at all points during the motion: 𝐸 = 𝐾 + 𝑈𝑔𝑟𝑎𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (if only gravity does work) A quantity that always has the same value is called a conserved quantity. When only the force of gravity does work, the total mechanical energy is constant— that is, it is conserved. This is our first example of the conservation of mechanical energy.  CONSERVATION OF MECHANICAL ENERGY A 0.145-kg baseball is thrown straight up, giving it an initial velocity of magnitude 20.0 m/s. Find how high it goes, ignoring air resistance. 𝐾1 + 𝑈𝑔𝑟𝑎𝑣,1 = 𝐾2 + 𝑈𝑔𝑟𝑎𝑣,2 𝑦1 = 0, 𝑈𝑔𝑟𝑎𝑣, 1 = 𝑚𝑔𝑦1 = 0, 𝑎𝑛𝑑 𝐾2 = ½ 𝑚𝑣22 = 0 𝐾1 = 𝑈𝑔𝑟𝑎𝑣,2 ½ 𝑚𝑣12 = 𝑚𝑔𝑦2 𝑣12 𝑦2 = 2𝑔 20.02 = 2×9.8 = 20.4𝑚  CONSERVATION OF MECHANICAL ENERGY WHEN FORCES OTHER THAN GRAVITY DOES WORK If other forces act on the body in addition to its weight then 𝐹𝑜𝑡ℎ𝑒𝑟 ≠ 0. For the pile driver, the force applied by the hoisting cable and the friction with the vertical guide rails are examples of forces that might be included in 𝐹𝑜𝑡ℎ𝑒𝑟. The total work Wtot by all forces is then the sum of Wgrav and the work done by 𝐹𝑜𝑡ℎ𝑒𝑟. 𝑊𝑡𝑜𝑡 = 𝑊𝑔𝑟𝑎𝑣 + 𝑊𝑜𝑡ℎ𝑒𝑟 Equating this to the change in kinetic energy, we have The work–energy theorem says that the total work done on the body equals the change in the body’s kinetic energy: 𝑊𝑡𝑜𝑡 = ∆𝐾 𝑊𝑔𝑟𝑎𝑣 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 − 𝐾1 But 𝑊𝑔𝑟𝑎𝑣 = 𝑈 𝑔𝑟𝑎𝑣, 1 − 𝑈 𝑔𝑟𝑎𝑣, 2 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 = ∆𝐸 𝑈 𝑔𝑟𝑎𝑣, 1 − 𝑈 𝑔𝑟𝑎𝑣, 2 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 − 𝐾1 The work done by all forces other than the 𝐾1 + 𝑈 𝑔𝑟𝑎𝑣, 1 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 + 𝑈 𝑔𝑟𝑎𝑣, 2 gravitational force equals the change in the total mechanical energy E  CONSERVATION OF MECHANICAL ENERGY WHEN FORCES OTHER THAN GRAVITY DOES WORK 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 = ∆𝐸 The work done by all forces other than the gravitational force equals the change in the total mechanical energy E  CONSERVATION OF MECHANICAL ENERGY WHEN FORCES OTHER THAN GRAVITY DOES WORK Suppose your hand moves upward by 0.50 m while you are throwing the ball of 0.145 kg. The ball leaves your hand with an upward velocity of 20.0 m/s. (a) Find the magnitude of the force (assumed constant) that your hand exerts on the ball. (b) Find the speed of the ball at a point 15.0 m above the point where it leaves your hand. Ignore air resistance. 𝐾1 + 𝑈 𝑔𝑟𝑎𝑣 , 1 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 + 𝑈 𝑔𝑟𝑎𝑣 , 2 1 0 + 0.145 × 9.8 × −0.50 + 𝑊𝑜𝑡ℎ𝑒𝑟 = × 0.145 × 20.0 2 +0 2 𝑊𝑜𝑡ℎ𝑒𝑟 = 29.7 𝐽 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐹 𝑦2 − 𝑦1 magnitude of the force (assumed constant) that your hand 𝑊 exerts on the ball, 𝐹 = 0𝑡ℎ𝑒𝑟 𝑦2−𝑦1 29.7 = = 59 𝑁 0.50  CONSERVATION OF MECHANICAL ENERGY WHEN FORCES OTHER THAN GRAVITY DOES WORK (b) Find the speed of the ball at a point 15.0 m above the point where it leaves your hand. Ignore air resistance 𝐾3 + 𝑈 𝑔𝑟𝑎𝑣 , 3 = 𝐾2 + 𝑈 𝑔𝑟𝑎𝑣 , 2 𝐾3 = 𝑈 𝑔𝑟𝑎𝑣 , 3 + 𝐾2 + 𝑈 𝑔𝑟𝑎𝑣 , 2 𝐾3 = −𝑈 𝑔𝑟𝑎𝑣 , 3 + 𝐾2 + 𝑈 𝑔𝑟𝑎𝑣 , 2 1 mv32 = − 0.145 × 9.8 × 15 + 29.7 + 0 = 7.7 𝐽 2 2𝐾3 v3 = ± 𝑚 7.7 = 2× = ±10𝑚/𝑠 0.145  GRAVITATIONAL POTENTIAL ENERGY FOR MOTION ALONG A CURVED PATH The body is acted on by the gravitational force 𝑤 = 𝑚𝑔 and possibly by other forces whose resultant is 𝐹𝑜𝑡ℎ𝑒𝑟. To find the work Wgrav done by the gravitational force during this displacement, we divide the path into small segments ∆𝑠. In terms of unit vectors, the force is 𝑤 = 𝑚𝑔 = 𝑚𝑔𝑗 and the displacement is ∆𝑠 = ∆𝑥 𝑖 + ∆𝑦𝑗 The work done by the gravitational force over this segment is the scalar product of the force and the displacement 𝑊𝑔𝑟𝑎𝑣 = 𝑤. ∆𝑠 = −𝑚𝑔𝑗. ∆𝑥 𝑖 + ∆𝑦𝑗 = −𝑚𝑔∆𝑦 = −𝑚𝑔 𝑦2 − 𝑦1 = 𝑚𝑔𝑦1 − 𝑚𝑔𝑦2 = 𝑈𝑔𝑟𝑎𝑣 , 1 − 𝑈𝑔𝑟𝑎𝑣, 2 So even if the path a body follows between two points is curved, the total work done by the gravitational force depends on only the difference in height between the two points of the path. This work is unaffected by any horizontal motion that may occur. So we can use the same expression for gravitational potential energy whether the body’s path is curved or straight.  ELASTIC POTENTIAL ENERGY In many situations we encounter potential energy that is not gravitational in nature. One example is a rubber-band slingshot. Work is done on the rubber band by the force that stretches it, that work is stored in the rubber band until you let it go. Then the rubber band gives kinetic energy to the projectile. We’ll describe The process of storing energy in a deformable body such as a spring or rubber band in terms of elastic potential energy. A body is called elastic if it returns to its original shape and size after being deformed.  ELASTIC POTENTIAL ENERGY Consider an ideal spring, but with its left end held stationary and its right end attached to a block with mass m that can move along the x-axis. The block is at x = 0 when the spring is neither stretched nor compressed. We move the block to one side, thereby stretching or compressing the spring, then let it go. As the block moves from a different position x1 to a different position x2 the work we must do on the spring to move one end from an elongation x1 to a different elongation x2 is where k is the force constant of the spring If we stretch the spring farther, we do positive work on the spring; if we let the spring relax while holding one end, we do negative work on it.  ELASTIC POTENTIAL ENERGY Now, from Newton’s third law the work done by the spring is just the negative of the work done on the spring. we find that in a displacement from x1 to x2 the spring does an amount of work Wel given by.  ELASTIC POTENTIAL ENERGY When a stretched spring is stretched farther, Wel is negative and Uel increases; more elastic potential energy is stored in the spring. When a stretched spring relaxes, x decreases, Wel is positive, and Uel decreases; the spring loses elastic potential energy. Uel is positive 1 for both positive and negative x values (𝑈𝑒𝑙 = 𝑘 𝑥 ) 2 2 The work–energy theorem says that the total work done on the body equals the change in the body’s kinetic energy: 𝑊𝑡𝑜𝑡 = ∆𝐾 = 𝐾2 − 𝐾1. If elastic force is the only force that acts, then 𝑊𝑡𝑜𝑡 = 𝑊𝑒𝑙 = 𝑈𝑒𝑙 , 1 − 𝑈𝑒𝑙 , 2 Putting these together, we get 𝐾2 − 𝐾1 = 𝑈𝑒𝑙 , 1 − 𝑈𝑒𝑙 , 2  CONSERVATION OF MECHANICAL ENERGY WHEN ELASTIC FORCE ALONE ACTS In this case the total mechanical energy E = K + Uel—the sum of kinetic and elastic potential energies—is conserved. An example of this is the motion of the block attached to a spring fixed at other end, provided the horizontal surface is frictionless so no force does work other than that exerted by the spring. For the Eq. to be strictly correct, the ideal spring that we’ve been discussing must also be massless. If the spring has mass, it also has kinetic energy as the coils of the spring move back and forth. We can ignore the kinetic energy of the spring if its mass is much less than the mass m of the body attached to the spring.  APPLICATION OF ELASTIC POTENTIAL ENERGY  CONSERVATION OF MECHANICAL ENERGY WHEN ELASTIC FORCE AND GRAVITATIONAL FORCES ACTS When we have both gravitational and elastic forces, such as a block attached to the lower end of a vertically hanging spring And if work is also done by other forces that cannot be described in terms of potential energy, such as the force of air resistance on a moving block? Then the total work is the sum of the work done by the gravitational force Wgrav, the work done by the elastic force Wel, and the work done by other forces Wother 𝑊𝑡𝑜𝑡 = 𝑊𝑔𝑟𝑎𝑣 + 𝑊𝑒𝑙 + 𝑊𝑜𝑡ℎ𝑒𝑟 The work–energy theorem says that the total work done on the body equals the change in the body’s kinetic energy: 𝑊𝑡𝑜𝑡 = ∆𝐾 𝑊𝑔𝑟𝑎𝑣 + 𝑊𝑒𝑙 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 − 𝐾1 But 𝑊𝑔𝑟𝑎𝑣 = 𝑈 𝑔𝑟𝑎𝑣 , 1 − 𝑈 𝑔𝑟𝑎𝑣 , 2 and 𝑊𝑒𝑙 = 𝑈𝑒𝑙, 1 − 𝑈𝑒𝑙 , 2 𝑈𝑔𝑟𝑎𝑣, 1 − 𝑈𝑔𝑟𝑎𝑣, 2 + 𝑈𝑒𝑙, 1 − 𝑈𝑒𝑙, 2 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 − 𝐾1 𝐾1 + 𝑈 𝑔𝑟𝑎𝑣 , 1 + 𝑈𝑒𝑙, 1 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 + 𝑈 𝑔𝑟𝑎𝑣, 2 + 𝑈𝑒𝑙, 2  CONSERVATION OF MECHANICAL ENERGY WHEN ELASTIC FORCE AND GRAVITATIONAL FORCES ACTS the most general statement of the relationship among kinetic energy, potential energy, and work done by other forces. It says: The work done by all forces other than the gravitational force or elastic force equals the change in the total mechanical energy E = K + U of the system. The “system” is made up of the body of mass m, the earth with which it interacts through the gravitational force, and the spring of force constant k. If Wother is positive, E = K + U increases; if Wother is negative, E decreases. If the gravitational and elastic forces are the only forces that do work on the body, then Wother = 0 and the total mechanical energy E = K + U is conserved.  CONSERVATION OF MECHANICAL ENERGY WHEN ELASTIC FORCE AND GRAVITATIONAL FORCES ACTS Trampoline jumping involves transformations among kinetic energy, elastic potential energy, and gravitational potential energy. As the jumper descends through the air from the high point of the bounce, gravitational potential energy Ugrav decreases and kinetic energy K increases. Once the jumper touches the trampoline, some of the mechanical energy goes into elastic potential energy Uel stored in the trampoline’s springs. At the lowest point of the trajectory (Ugrav is minimum), the jumper comes to a momentary halt (K = 0) and the springs are maximally stretched (Uel is maximum). The springs then convert their energy back into K and Ugrav, propelling the jumper upward