Potential energy Basic energy Lecture 10 Mechanical Energy Conservation; Power
ACT: Zero net work The system of pulleys shown below is used to lift a bag of mass M at constant speed a distance h from the floor. What is the work done by the person? A. Mgh B. ½Mgh C. 2Mgh h M
Work done by gravity A block of mass m is lifted from the floor (A) to a table (B) through two different trajectories. Find the work done by gravity. Δy θ B Δr W = mg (Δ r 1 cos θ 1 Δ r 2 cos θ 2 Δ r 3 cos θ 3 ) = mg (Δ y Δr 1 + y 2 + Δ y 3 ) 3 θ 2 θ 3 W = mg Δ rcosθ = m g Δ y = mg Δ y A θ 1 mg Δr 1 Δr 2 Work by gravity does not depend on the path; only depends on change of height.
Gravitational potential energy The work done by gravity does not depend on the path, it only depends on the vertical displacement Δy, or on the initial and final y : W = m g Δ y We can ALWAYS write this work as (minus) the change in some function PE (r) that depends on position (not on path): W = (PE f PE i ) = ΔPE PE = potential energy Gravitational potential energy: PE G = mgy
Elastic Potential Energy (Spring) What is the work done by a spring as the tip is pulled from x 1 to x 2? ( W by spring = 1 2 k x 2 1 2 2 k x ) 2 1 x 1 F 1 = -kx 1 F 2 = -kx 2 This can also be written as (minus) the difference of a potential function at the initial and final points x 1 and x 2 : W by spring = [PE (x 2 ) PE (x 1 )] x 2 Elastic potential energy: PE spring = 1 2 k x 2
Can work always be written in terms of a potential energy change? A NO! Example: A box is dragged along a rough horizontal surface through two paths between same two points: D W W friction,ad = -df = - k df friction,abcd 3 k Does not depend on initial and final points only. B C The work done by friction CANNOT be written as a potential difference.
Conservative and non-conservative forces The work done by a conservative force does not depend on the trajectory. A potential energy function can be defined. Examples: Gravity, spring Non-conservative force = force that is not conservative. The work done by a non-conservative force depends on the trajectory. A potential energy function cannot be defined. Examples: Kinetic friction
Conservation of Mechanical Energy In a system where both conservative and nonconservative forces are doing work, we can rewrite the WKE theorem: W net = ΔKE ΔPE + W nc = Δ KE W nc = ΔKE + ΔPE W net = Δ PE + W nc Definition of Mechanical Energy: E = KE + PE Δ E = W nc When no non-coservative force is present: ΔE = constant
Example: Free fall A ball is dropped from a height h. If the initial speed is 0 and we ignore air resistance, what is the speed of the ball as it hits the ground?
Example: Free fall A ball is dropped from a height h. If the initial speed is 0 and we ignore air resistance, what is the speed of the ball as it hits the ground? We can use kinematics or the WKE theorem or conservation of energy. Δr mg WKE Work done by gravity: mgh Conservation of energy The only force doing work is gravity, so mechanical energy is conserved. E initial = E final W mgh v = ΔKE = 1 2 mv2 = 2gh KE initial + PE initital = KE final + PE final 1 2 0 + mgh = + 0 2 mv 1 2 mgh = mv 2 v = 2gh Choice: PE = 0 at ground level
ACT: Up an incline A box of mass m and initial speed v 0 = 10 m/s moves up a frictionless incline angled 30. How high does the box go before it begins sliding down? A. 2 m B. 5 m C. 10 m
ACT: Up an incline A box of mass m and initial speed v 0 = 10 m/s moves up a frictionless incline angled 30. How high does the box go before it begins sliding down? E = KE + PE A. 2 m E A = 1 2 m v 0 2 + 0 B. 5 m C. 10 m E B = 0 + mgh E A = E B 1 2 m v 02 = mgh Only gravity does work (the normal is perpendicular to the motion), so mechanical energy is conserved. h = 2 2 v 0 (10 m/s) 2 g = 2 (9.8 m/s 2 ) = 5.1 m
The really nice thing is, we can apply the same thing to any incline : Turnaround point: where KE = 0 h E KE PE v = 0 E KE PE DEMO: Wavy track E KE PE
Cool Example: Loop-the-loop A cart is released from height h in a roller coaster with a loop of radius R. What is the minimum h to keep the cart on the track? h A. 1.5R B. 2.0R C. 2.5R D. 3.0R E. 4.0R R
Cool Example: Loop-the-loop A cart is released from height h in a roller coaster with a loop of radius R. What is the minimum h to keep the cart on the track? h Impossible, h must be at least 2R A. 1.5R B. 2.0R C. 2.5R D. 3.0R E. 4.0R R
h Point B is the toughest point. What is the speed there? Aaaah!!!! A EA = EB 1 mgh + 0 = mg2r + mvb 2 vb = 2 g( h - 2 R ) B R 2 (Eqn. 1)
In order not to fall (ie, to keep the circular trajectory), the forces at B must provide the appropriate radial acceleration: + = 2 v B mg N m R Aaaah!!!! A The minimum velocity is fixed by N = 0: 2 vb,min mg = m vb,min = gr R B (Eqn. 2) h mg N by track R
Let us put equations 1 and 2 together: vb = 2 g( h - 2 R ) vb,min = gr The minimum height is given by: gr = 2 g( h - 2 R ) R = 2h - 4R min min h min 5 = R Answer C 2
Power Power is the rate at which work is done: P = work energy transferred/transformed = time time In the SI system, the units of power are watts: 1 W = 1 J/s The difference between walking and running up these stairs is power the change in gravitational potential energy is the same.
P = work time = F Δ x Δ t =F v P engine > P gravity + P N + P R Car speeds up. P engine < P gravity + P N + P R Car slows down. P engine = P gravity + P N + P R Final speed equals initial speed (total power is zero) P gravity = mg v x sin10 P N = F R v y = 0 P R = F R v x P engine = F v x