Chapter 7: Potential energy and energy conservation Two types of Potential energy gravitational and elastic potential energy Conservation of total mechanical energy When What: Kinetic energy+potential energy Sometimes total mechanical energy is not conserved Conservative forces VS nonconservative forces The law of energy conservation Force can be determined if the potential energy is known
Gravitational potential energy When one drops a basketball, it hits the ground with certain kinetic energy. Where does it come from? W grav = Fs = w(y 1 y 2 ) = mgy 1 mgy 2 U grav = mgy Gravitational force did the work! Gravitational potential energy is transformed into kinetic energy
Conservation of energy (Mechanical) Energy is conserved, and we can prove it! Gravitational potential energy kinetic energy W tot = ΔK (where did this come from?) W tot = W grav = U 1 U 2 = ΔU ΔK = ΔU or K 2 K 1 = U 1 U 2 E K 1 +U 1 = K 2 +U 2 If only gravity does work, sum of kinetic energy and gravitational potential energy remain constant! Does zero height matter? It s always the difference of height!
Application of energy conservation. We can determine how high a baseball can reach: Does the mass matter? Does free fall motion formula gives the same result? E 1 = 1 2 mv 2 1 E 2 = mgy 2 E 1 = E 2 1 2 mv 2 1 = mgy 2 y 2 = v 1 2 2g
Other forces doing work will change the total energy 1 2 mv 2 1 + mgy 1 +W other = 1 2 mv 2 2 + mgy 2 When throwing a baseball, we apply the other force before releasing it. What is v 3 and the force we applied on it? F(y 2 y 1 ) = E 2 E 1 E 2 E 1 = 1 2 mv 2 2 mgy 1 F =? E 2 = E 3 1 2 mv 2 2 + mgy 2 = 1 2 mv 3 2 + mgy 3 Can v 3 be negative?
Q7.1 A piece of fruit falls straight down. As it falls, A. the gravitational force does positive work on it and the gravitational potential energy increases. B. the gravitational force does positive work on it and the gravitational potential energy decreases. C. the gravitational force does negative work on it and the gravitational potential energy increases. D. the gravitational force does negative work on it and the gravitational potential energy decreases.
Q7.2 You toss a 0.150-kg baseball straight upward so that it leaves your hand moving at 20.0 m/s. The ball reaches a maximum height y 2. What is the speed of the ball when it is at a height of y 2 /2? Ignore air resistance. A. 10.0 m/s B. less than 10.0 m/s but more than zero C. more than 10.0 m/s v 2 = 0 v 1 = 20.0 m/s m = 0.150 kg D. not enough information given to decide y 2 y 1 = 0
Work and energy along a curved path For gravitational energy, only the height matters, not the path! For a curved path, U grav is solely dependent on the height!
Q7.3 As a rock slides from A to B along the inside of this frictionless hemispherical bowl, mechanical energy is conserved. Why? (Ignore air resistance.) A. The bowl is hemispherical. B. The normal force is balanced by centrifugal force. C. The normal force is balanced by centripetal force. D. The normal force acts perpendicular to the bowl s surface. E. The rock s acceleration is perpendicular to the bowl s surface.
Q7.4 The two ramps shown are both frictionless. The heights y 1 and y 2 are the same for each ramp. A block of mass m is released from rest at the left-hand end of each ramp. Which block arrives at the righthand end with the greater speed? A. the block on the curved track B. the block on the straight track C. Both blocks arrive at the right-hand end with the same speed. D. The answer depends on the shape of the curved track.
Consider projectile motion using energetics Knowing that the initial speeds are the same (v 0 ), what is the speed at height=h? Does it matter at what angle it is released? E 1 = E 2 1 2 mv 2 1 = mgh + 1 2 mv 2 2 The direction of v 1 or v 2 has no effect of v 2 magnitude!
What s the speed in a vertical circle? K 1 =? U 1 =? K 2 =? U 2 =? A particle moving in a full vertical frictionless circle. It is released from the bottom at the speed of v 0 = What is the speed on the top? 68gR
Real work always has friction, though If the skater reaches the bottom at 6.00m/s instead of 7.67m/s, due to friction? How much work did friction do? Where did those missing energy go? One could also use Newton s second law. But it is difficult to apply. Do you want the easy way? The energy approach is the SHORTCUT you should take!
The energy of a crate on an inclined plane Not thinking about friction, man pushes a 12kg crate at 5.0m/s, only to see that it slid 1.6m, and slid back down. What is the magnitude of the friction? How fast will the crate be at the bottom? Is E 1 -E 2 =E 3 -E 2?
Work and energy in the motion of a mass on a spring Work done on elastic (spring force) is stored (in the spring), as elastic potential energy. U el = 1 2 kx 2 W el = W = 1 2 kx 2 1 1 2 kx 2 2 = ΔU el If only elastic force does work: 1 2 mv 2 1 + 1 2 kx 2 1 = 1 2 mv 2 2 + 1 2 kx 22
Potential energy curve It s symmetric! And parabolic! Both stretching and compressing the spring will increase the elastic potential energy! When do we have minimum elastic potential energy? U el = 1 2 kx 2
Bungee jumping: two potential energies in play! Gravitational energy kinetic energy+elastic potential energy elastic potential energy Will this go on forever?
Motion with elastic potential energy What is v 2x? If we apply F=+0.610N, and move it to x 1 =0.1m, what is v 1x, and what is v 2x when it comes back to x 2, and x 0 =0? k=?
Q7.5 A block is released from rest on a frictionless incline as shown. When the moving block is in contact with the spring and compressing it, what is happening to the gravitational potential energy U grav and the elastic potential energy U el? A. U grav and U el are both increasing. B. U grav and U el are both decreasing. C. U grav is increasing, U el is decreasing. D. U grav is decreasing, U el is increasing. E. The answer depends on how the block s speed is changing.
Worst case scenario: two potential energies and friction A) In order to stop a falling elevator using friction and a cushioning spring, how strong should the spring be (k=?) W f = E 2 E 1 = f ( y 2 ) E 2 = K 2 +U 2 = 0 + ( 1 2 ky 2 2 + mgy 2 ) E 1 = K 1 = 1 2 mv 2 1 k = mv 1 2 + 2(mg f )( y 2 ) y 2 2 B) What if there is no friction available? y 1 =0 y 2 = -2.0m Friction must be larger than k(-y 2 )-mg? Otherwise what would happen?
Conservative forces The work by a conservative force like gravity does not depend on the path your hiking team chooses, only how high you climb.
Force as the derivative of potential energy W = ΔU W = FΔx FΔx = ΔU F = ΔU du(x) F(x) = Δx dx and F (x, y,z) = U(x, y,z) = x ˆ i + y ˆ j + z and U(x, y,z) F x = x U(x, y,z) F y = y F z = U(x, y,z) z ˆ k
Energy diagrams give us insight Figure 7.23 shows the situation and the resulting energy diagram. Information may be discerned about limits and zeros for the physical properties involved.
The potential energy curve for motion of a particle Refer the potential energy function and its corresponding components of force.
Q7.8 The graph shows the potential energy U for a particle that moves along the x-axis. At which of the labeled x-coordinates is there zero force on the particle? A. at x = a and x = c B. at x = b only C. at x = d only D. at x = b and d E. misleading question there is a force at all values of x.
Q7.10 The graph shows a conservative force F x as a function of x in the vicinity of x = a. As the graph shows, F x = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? 0 F x a x A. U = 0 at x = a B. U is a maximum at x = a. C. U is a minimum at x = a. D. U is neither a minimum or a maximum at x = a, and its value at x = a need not be zero.
Q7.11 The graph shows a conservative force F x as a function of x in the vicinity of x = a. As the graph shows, F x > 0 and df x /dx < 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? 0 F x a x A. du/dx > 0 at x = a B. du/dx < 0 at x = a C. du/dx = 0 at x = a D. Any of the above could be correct.
Summary of CH7: Potential energy Two kinds of potential energies: Gravitational: U gr = mgh Elastic: U el = 1 2 kx 2 F x (x) = du dx Corresponding forces: w = mg F = kx Mechanical Energy conservation: With other forces than gr and el: K 1 +U 1 = K 2 +U 2 K 1 +U 1 +W other = K 2 +U 2 (Friction Internal Energy) Law of energy conservation: From potential energy to force: (required) (not-required) K 1 +U 1 ΔU int = K 2 +U 2 (ΔK + ΔU + ΔU int = 0) F x (x) = du dx F = U = ( U i x ˆ + U x ˆ j + U x ˆ k )