Chapter 5: Energy Energy is one of the most important concepts in the world of science. Common forms of Energy Mechanical Chemical Thermal Electromagnetic Nuclear One form of energy can be converted to another form, but the total amount of energy in the universe never changes. Work W ( F cosθ ) Δx F is the magnitude of the force Δ x is the magnitude of the object s displacement θ is r the angle between r F and Δx Work provides a link between force and energy.
Work This gives no information about the time it took for the displacement to occur the velocity or acceleration of the object Work is a scalar quantity, can be positive, negtive, or zero. SI unit of Work Newton meter = Joule N m = J J = kg m / s v f v m v i f What changes when Work is done? A non-zero net force results in acceleration of an object, thus change of velocity. But, change of what quantity can be characterized by Work? FΔx W = a Δx = = m m mv = W i
Kinetic Energy The kinetic energy KE of an object with mass m and speed v is given by KE = mv The Work-Kinetic Energy Theorem W= KE f KE i = ΔKE = m v f m vi Example: Downhill Skiing A 58-kg skier is coasting down a 5 o slope. A kinetic frictional force of magnitude f k = 70N opposes her motion. Near the top of the slope, the skier s speed is v i = 3.6 m/s. Ignore air resistance, determine the speed v f at a point that is displaced 57m downhill. Work done by gravitational force is W G = 58 * 9.8 * 57 * sin5 o =.37x0 4 J Work done by frictional force is W F = - 70 N * 57 m = - 4.0 x 0 3 J Total work = W G + W F = 9.7x 0 3 J = ΔKE = 0.5*58*v f - 0.5*58*3.6 Therefore, v f = 9 m/s
A force is a conservative force when the work it does on an object moving between two points is independent of the path between the two points. Conservative Forces Examples of Conservative Forces: Gravitational force Elastic spring force Electrostatic force Gravitational Potential Energy W gravity = mg (y i y f ) The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured by the height y of the object relative to an arbitrary zero level: PE = mgy W g = PE i -PE f SI Unit of PE is joule (J). The choice for the location of zero potential energy is arbitrary, because only change in potential energy has physical consequence.
Mechanical Energy ΔKE = W = W nc + W g = W nc + (PE i PE f ) KE f KE i = ΔKE W = ( KE KE ) + ( PE PE ) nc f i f i E = KE + PE W nc = E f E i When there s no non-conservative force, W nc =0, and E f = E i Examples of Conserv. of Mech. Energy Three balls thrown with same initial speed. Rank the speeds of the balls as they reach the ground. Projectile Motion A motorcyclist is trying to leap across the canyon by driving horizontally off the cliff at a speed of 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side.
Work done by a varying force W = ( Fcosθ ) Δs + ( Fcosθ ) Δs + Work done by a varying force is the area under the curve of F vs x. Potential Energy POTENTIAL ENERGY at point A is the negative of the work done by a conservative force from point O to point A. is the work that need to be done against a conservative force to bring an object at rest from point O to point A. is the energy stored in a conservative force field is the potential (ability, tendency,..) of a conservative force field to do work
Hooke s Law: Springs and Hooke s Law The restoring force of an ideal spring is F = - k x where k is the spring constant and x is the displacement of the spring from its unstrained length. The minus sign indicates that the restoring force always points in a direction opposite to the displacement of the spring. Unit of k: N/m Elastic spring force is a conservative force. Potential Energy of Compressed (of Stretched) Spring Spring potential energy is the work required to compress or stretch a spring from its equilibrium position to a final position x. F = k x The work is the area under that curve: W = kx / Therefore, the elastic potential energy of a stretched or compressed spring is PE elastic = kx / Note: To consider the potential energy of the spring, we CANNOT arbitrarily define where x=0 is.
Total Mechanical Energy Involving Springs KE f KE i = W nc + W g + W s = W nc (PE gf PE gi ) - (PE sf PE si ) W nc = (KE f KE i) ) + (PE gf PE gi ) + (PE sf PE si ) = (KE+PE g +PE s ) f -(KE+PE g +PE s ) i Total mechanical energy: E Total = KE + PE g + PE s = mv + mgy + If W nc = 0, E total is conserved. kx Example: Mechanical Energy Involving Springs Total mechanical energy E Total = mv + mgy + kx The launching mechanism of a toy gun consists of a spring of unknown spring constant, as shown in. If the spring is compressed a distance of 0.0 m and the gun fired vertically as shown, the gun can launch a 0.0-g projectile from rest to a maximum height of 0.0 m above the starting point of the projectile. Neglecting all resistive forces, determine (a) the spring constant and (b) the speed of the projectile as it moves through the equilibrium position of the spring (where x = 0).
Power Definition of Average Power Average power P is the average rate at which work W is done, and it is obtained by dividing W by the time t required to perform the work P = Work Time = W Δt If the force is constant and points in the same direction as the displacement, the average power can be written as W F Δx P = = = Δt Δt SI Unit of Power: joule/s = watt (W) F v ft lb hp = 550 = 746 W s Example Problems: Chap. 5 Two blocks, A and B (with mass 50 kg and 00 kg, respectively), are connected by a string. The pulley is frictionless and of negligible mass. The coefficient of kinetic friction between block A and the incline is μ k = 0.5. Determine the change in the kinetic energy of block A as it moves from C to D, a distance of 0 m up the incline if the system starts from rest. C D A child slides without friction from a height h along a curved water slide. She is launched from a height h/5 into the pool. Determine her maximum airborne height y in terms of h and θ.
Review of Chapter 5 Work W = (F cosθ)s Kinetic energy = mv / Work done by net external forces = ΔKE Conservative and non-conservative forces Gravitational potential energy = mgy Hooke s law and springs F = -kx Elastic potential energy / kx Total mechanical energy = KE + PE Principle of conservation of mechanical energy Average power Principle of conservation of energy