Conservation of Energy and Momentum

Transcription

1 Conservation of Energy and Momentum

2 Three criteria for Work There must be a force. There must be a displacement, d. The force must have a component parallel to the displacement. Work, W = F x d, W = Fd cos θ. F F x

3 If a Force does not affect displacement, it does NO WORK. F A F g The Earth exerts a force on the case, but does no work. The man exerts a horizontal force & does work.

4 Ex :Work on a backpack Determine the work a hiker must do on a 5.0kg backpack to carry it up a hill of height h=0.0m. d θ h h = d cosθ Soln: In the vertical direction, if up is positive, ΣF y = ma y If mg is the force of gravity down, and F h is the hiker s force up, then ΣF y = F h - mg = 0 (assuming negligible acceleration) so F h = mg = 47N. Work done by the hiker, W h = F h (d cos θ) = F h h = mgh = (47N )(0.0m) = 470 J

5 Does Earth do Work on the Moon? last (third)quarter waning Moon SUN crescent moon orbit`s earth v gibbous moon earth F g new moon full moon crescent gibbous moon waxing Moon first quarter The angle between the Force and instantaneous displacement θ is 90. Cos 90=0, therefore NO WORK is done by gravity. (W=Fdcosθ)

6 Kinetic Energy Regardless of the types of Energy, in any process it is CRUCIAL to understand the total energy remains the same after the process as before! Energy traditionally is defined as the ability to do work or cause change. (change = acceleration) Kinetic Energy is energy of motion. A moving object has the ability to do work. KE = ½ mv and is measured in Joules, J.

7 Do Work! Doing work on an object causes a change in the object s energy More work = more change. W net = KE KE so W net = ΔKE The net work done on an object is equal to the change in its kinetic energy. This is the Work-Energy principle.

8 Work to stop a car An automobile traveling 60 km/h can brake to stop within a distance of 0m. If the car is going twice as fast, what is its stopping distance?

9 Solution The stopping force, F is about constant. The work needed to stop the car, Fd, is proportional to the stopping distance. Using the Work-Energy theorem we note that F and d are in opposite directions: W net = Fd(cosθ) = Fdcos80= -Fd W net = ΔKE = 0 ½ mv Since force & mass are constant, stopping distance increases as v If v is doubled, stopping distance is as great or 0m(4) = 80m.

10 Your Turn to Practice Please do Ch 6 Rev p 74 #s -6 Pg 75 #s 7-0

11 Potential Energy Energy stored in an object due to its position or condition is Potential Energy. Common types of Potential energy include: Gravitational PE energy due to position above Earth. Elastic PE- energy stored due to work being done on an object and the object restoring its original shape.

12 Gravitational PE Gravitational Potential Energy results from lifting an object to a height, h above some reference point (like the ground). PE grav =mgy (where y is the height above y 0 = 0m). Work done to lift a mass from y to y is equal to the change in PE from y to y. Just like KE, though we may not know the original amount of energy an object has, if we do WORK on it, we know the change in energy. W = mg (y y ) = ΔPE.

13 It is important to note, CHANGES in PE depend only on vertical height, h, and not on the path taken to get there Recall the hiker and backpack? How about the Pyramids of Ancient Egypt? Does the horizontal part of stairs play a role in your PE as you climb them?

14 Elastic Potential Energy If you push (or pull) on a spring (you do work to compress or stretch it), the spring can store energy and can do work when it is released. To hold a spring, stretched or compressed, a distance, X from its normal un-stretched length, requires a force, F p that is directly proportional to X. F p = kx where k is a spring constant based on stiffness of the spring material.

15 Hooke s Law A stretched or compressed spring, wants to restore itself in the opposite direction. This restoring force is the spring s force. This is Hooke s Law or the Spring equation: F s = - kx, where k is the spring constant and the negative sign is because the spring exerts the force in a direction opposite the displacement. x m F

16 Compressing or Stretching a Spring Initially at Rest: Two forces are always present: the outside force F out ON spring and the reaction force F s BY the spring. x m Compressing x m Stretching Compression: F out does positive work and F s does negative work (see figure). Stretching: F out does positive work and F s does negative work (see figure).

17 General Case for Springs: If the initial displacement is not zero, the work done is given by: Work kx kx F x x x x m m

18 Stretch a spring As you pull on a spring, the Force varies with the stretch (to stretch twice as far does NOT require twice the force!) Potential energy of a spring is found considering Work done on it to stretch it. Consider AVERAGE FORCE to stretch a spring from 0 to x from F p = kx, we get F W F 0 p x kx Elastic Potential Energy = ½ kx (x = displacement stretched or compressed from normal equilibrium position.) kx kx x kx

19 Conservation of Mechanical Energy E is the total mechanical energy of a system. E is the sum of kinetic and potential energies at any moment. E = KE + PE KE + PE = KE + PE E = E = constant OR: Therefore: ΔPE = -ΔKE or if KE increases, then PE must decrease.

20 Problem solving with Energy conservation A rock at height h is dropped from rest. (KE = 0). As it falls, PE decreases, but it gains speed (KE increases = to the change of PE) At any point, the total mechanical energy is given by: E = KE + PE = ½ mv + mgy where v is the velocity at height y 3

21 Total Mechanical energy at different points of fall must be equal. mv mgy mv mgy

22 Conservation of a spring A dart of mass 0.00kg is pressed against the spring of a toy dart gun. The spring, k = 50N/m, is compressed 6.0cm and released. If the dart detaches from the spring when the spring reaches equilibrium, what speed does the dart acquire?

23 mv kx mv kx In the Horizontal direction, the only force on the dart is the force exerted by the spring. Vertically, gravity is balanced by the normal force on the gun barrel. (After the dart leaves the gun, it follows the projectile path under gravity) Use the above equation w position being at the maximum compression of the spring, so v = 0 (dart not released yet) and x = m. Point we choose to be the instant the dart flies off the end of the spring, so x = 0 and we want to find v. We can rewrite the equation: 0 v v kx kx m v mv 0 (50N / m)(0.06m) (0.00kg) 3.0m / s 9.0 m s

24 Law of Conservation of Energy The total energy is neither increased nor decreased in any process. Energy can be transformed from one form to another, and transferred from one body to another, but the total amount remains constant.

25 Power Power is the rate at which work is done, or the rate at which energy is transformed. work energytransformed J P Watt time time s Another unit of power is the horsepower (hp). hp = 746 W.

26 Your turn to Practice Please do Ch 6 Rev p 75 #s,, 9, 30, 3 P 77 #s 58, 59, 60.