Rutgers University Department of Physics & Astronomy. 01:750:271 Honors Physics I Fall Lecture 10. Home Page. Title Page. Page 1 of 37.

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1 Rutgers University Department of Physics & Astronomy 01:750:271 Honors Physics I Fall 2015 Lecture 10 Page 1 of 37

2 Midterm I summary Average: Page 2 of 37

3 7. Kinetic energy and work Kinetic Energy: energy associated to the motion of an object K = 1 2 mv2 Page 3 of 37

4 Work done by an applied force Work done by the force F W = F dcos φ = F d d = ( x)î displacement vector Page 4 of 37

5 Work-Kinetic Energy Theorem K = W net K f = K i + W net Page 5 of 37 W net = W = F d = ( F ) d = Fnet d

6 Variable force, curved trajectory W = dw = F d s trajectory trajectory d s = vdt infinitesimal displacement F Page 6 of 37 ds

7 Example: A ball tied at the end of a string of length r moves on a circular trajectory under an applied force F = F ĵ. F net = F + T F q F net d s = F d s = F dscosθ ds q T mv 2 W = π/2 0 = F r 2 = mv F r v = F cosθ rdθ π/2 0 cosθ dθ = F r ( sin(π/2) sin(0)) = F r v F r m Page 7 of 37

8 Work done by a spring force Hooke s Law F s = k d always opposed to displacement (restoring force) Page 8 of 37 k > 0 spring constant

9 W s = = xf x i xf x i = ( k) F x dx kxdx xf x i xdx = ( k/2) ( x 2 f x2 i ) Page 9 of 37 W s = 1 2 kx2 i 1 2 kx2 f

10 Example: an object of mass m slides across a horizontal frictionless surface with speed v. It then runs into and compresses a spring of spring constant k. When the object is momentarily stopped by the spring, by what distance d is the spring compressed? Page 10 of 37

11 Total work done by the spring force: W s = kx2 i 2 kx2 f 2 = kd2 2 Work-kinetic energy theorem W s = K f K i = mv2 2 Page 11 of 37 mv 2 2 = kd2 2 d = v m k

12 Power: time rate at which work is done by a force. If a force does an amount of work W in an amount of time t, the average power during that time interval is: P average = W t The instantaneous power P is the instantaneous time rate of doing work Page 12 of 37 P = dw dt dw = F d s = F vdt P = F v

13 Units: Watt 1 Watt = 1 W = 1 J/s Page 13 of 37

14 8. Potential energy. Conservation of energy Potential energy: energy associated with the configuration of a system of objects that exert forces on one another. can be converted into kinetic energy by allowing the system to evolve freely Page 14 of 37

15 Page 15 of 37 Elastic potential energy Gravitational potential energy

16 Work and potential energy First part of motion: W Fg = K < 0, K energy transferred from kinetic energy to gravitational potential energy. Second part of motion: W Fg = K > 0, K energy transferred from gravitational potential energy to kinetic energy. Page 16 of 37

17 First part of motion: W Fs = K < 0, K energy transferred from kinetic energy to elastic potential energy. Second part of motion: W Fs = K > 0, K energy transferred from elastic potential energy to kinetic energy. Page 17 of 37

18 First part of motion: W Fg = mg(y 0 y max ) K = mv2 0 2 Constant acceleration model: mv = mg(y max y 0 ) Second part of motion: W Fg = mg(y max y 0 ) K = mv2 2 Constant acceleration model: mv 2 2 = mg(y max y 0 ) Page 18 of 37

19 First part of motion: W Fs = kx2 max 2 W Fs = K = mv2 0 2 mv0 2 2 = kx2 max 2 Second part of motion: Page 19 of 37 W Fs = kx2 max 2 W Fs = K = mv2 2 mv 2 2 = kx2 max 2

20 Note that in both examples examples W 1 st part = W 2 nd part Gravitational force: (mgy) = W Fg K + mgy = constant Elastic force: Page 20 of 37 (kx 2 /2) = W s K + kx2 2 = constant

21 Naturally led to: Gravitational potential energy: U g = mgy Elastic potential energy: Energy conservation: U s = kx2 2 Page 21 of 37 K + U g = constant K + U s = constant

22 i-clicker Which of the following statements is true? h a q a h b q b A) W (a) B) W (a) C) W (a) > W (b) < W (b) = W (b) D) none of the above. Page 22 of 37

23 Answer Which of the following statements is true? h a q a h b q b A) W (a) B) W (a) C) W (a) > W (b) < W (b) = W (b) D) none of the above. Page 23 of 37

24 W = d = x x x = mgsinθ F gy q F g F gx x = h sinθ W = mgh Page 24 of 37

25 i-clicker Which of the following statements is true? h a h b A) W (a) B) W (a) C) W (a) > W (b) < W (b) = W (b) D) none of the above. Page 25 of 37

26 Answer Which of the following statements is true? h a h b A) W (a) B) W (a) C) W (a) > W (b) < W (b) = W (b) D) none of the above. Page 26 of 37

27 W g = d s d s = mgds y ds y F g ds x ds W g = h 0 = mg = mgh. mgds y h 0 W = mgh ds y Page 27 of 37

28 Conservative Forces The work done by the force depends only on the initial and final position of the object, not on the path in between. The net work done by a conservative force on a particle moving around any closed path is zero. Page 28 of 37

29 Page 29 of 37 Consequence: when the configuration change is reversed the work changes sign: W a b = W b a

30 Examples: gravitational force, elastic force Potential energy for conservative forces: define U such that: Note: U = U f U i = W i f W i f is path independent, hence this is a consistent relation Choosing U 0 = 0 for some reference configuration: U a = W 0 a Page 30 of 37

31 Gravitational potential energy U g = mg(y f y i ) Reference configuration: ground level U g (0) = 0 U = mgy Elastic potential energy U s = k 2 (x2 f x2 i ) Page 31 of 37 Reference configuration: relaxed spring U s (0) = 0 U s = kx2 2

32 Conservation of Mechanical Energy In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy E mec of the system, cannot change. Page 32 of 37 Conservative forces, isolated system U + K = constant

33 i-clicker A block of mass m slides down a curved slope as shwon below. What is the final speed of the block? A) v = 2gy 1 B) v = 2gy 2 C) v = 2g(y 1 y 2 ) D) none of the above Page 33 of 37

34 Answer A block of mass m slides down a frictionless curved slope as shown below. What is the final speed of the block? A) v = 2gy 1 B) v = 2gy 2 C) v = 2g(y 1 y 2 ) D) none of the above Page 34 of 37 Energy conservation: mgy 1 = mgy 2 + mv 2 /2 v = 2g(y 1 y 2 )

35 Non-conservative (dissipative) forces: W depends on the path There is no potential energy U associated to a configuration such that U = W Page 35 of 37 Examples: kinetic friction, drag

36 Example: A u0 (1) B u Suppose an object is launched from A to B on a rough horizontal surface with kinetic friction coefficient µ k v 0 (1) along a straight line (2) on a circular trajectory (tied to a string) Page 36 of 37 A (2) B v W (1) A B = W (2) B A?

37 W (1) A B = µ kmgd AB W (2) A B = B A f k d s = π 2 µ kmgd AB In conclusion: W (1) A B W (2) B A Page 37 of 37