Physics 1A Lecture 6B. "If the only tool you have is a hammer, every problem looks like a nail. --Abraham Maslow

Transcription

1 Physics 1A Lecture 6B "If the only tool you have is a hammer, every problem looks like a nail. --Abraham Maslow

2 Work Let s assume a constant force F acts on a rolling ball in a trough at an angle θ over displacement Δx +y +x F θ v ox Δx Looking at the motion in the x-direction: v x Turning to the kinematics equations:

3 Work We want to link motion and force, so solve for a x : Plugging back into 2nd Law (only one force acting along x-direction): Dividing by displacement gives: This is Work- Energy Theorem:

4 Conceptual Question Two balls are dropped from the same height from the roof of a building. One ball has twice the mass as the other. Air resistance is negligible for this question. Just before hitting the ground, the heavier ball has: A) one-quarter the kinetic energy of the lighter ball. B) one-half the kinetic energy of the lighter ball. C) the same kinetic energy as the lighter ball. D) twice the kinetic energy of the lighter ball. E) four times the kinetic energy of the lighter ball.

5 Kinetic Energy Since both balls are dropped from the same height, they will both attain the same velocity when they reach the ground. (v 2 = v o 2 + 2aΔx) So, v heavy = v light = v at the bottom. But the heavier ball is twice the mass of the lighter ball. m heavy = 2m light So, KE light = (1/2)m light v 2 And, KE heavy = (1/2)m heavy v 2 = (1/2)(2m light )v 2 KE heavy = 2 KE light

6 Power Work and energy are not explicitly time dependent. It could take you 10 seconds to do 1 Joule of work or 10 hours. Work will never tell you how long it would take. But you can turn to average power, P avg, which is the rate at which work is done. If you want to turn to instantaneous power you get: P = lim Δt 0 W Δt = dw dt

7 Power The SI unit for power is: [1 Joule/sec] = [1 Watt]. Another unit that is commonly used is horsepower, hp, where: 1hp = 746W. If we have a constant force in one dimension we can write: P = dw dt = d ( F x ) x dt d( x) = F x dt P = F xv x If you want to extend this to more dimensions, just change to a scalar product: P = F v

8 Potential Energy Another type of energy is potential energy. Kinetic energy quantifies motion. Potential energy (PE or U) is the amount of stored energy you have that can perform work or be transferred to kinetic energy. Potential Energy is also measured in Joules. Potential Energy is also a state variable, it only depends on the initial and final values (it is not path dependent).

9 Potential Energy Work can also go into changing potential energy. If kinetic energy is constant, then we can say that between two points A and B: ΔU = W For the gravitational force: ΔU = y f y i ( mg) ˆ j dyˆ j ΔU = ΔU = B A B A F dr F g dr ΔU = mg( y f y ) i = mgδy Gravitational Potential Energy

10 Conservative Forces A conservative force is one that performs work that transfers energy between useable systems and is reversible. For example, if I drop an apple, the gravitational force performs work on the apple. Gravitational force transfers PE in the apple to KE in the apple. It is a conservative force, energy is not lost to other systems. A non-conservative force is one that transfers energy out of a particular system and is not reversible.

11 Conservative Forces How can you tell, in general, if a force is conservative or not? The easiest way is to have the force act over a closed path. If the force is conservative: W 1 + W 2 = 0 If the force in nonconservative: W 1 + W 2 0 In other words, the work due to a conservative force is path independent.

12 Conservative Forces When the total work done by a force F acting as an object moves over any closed loop is zero, then the force is conservative. Mathematically, we say: F d r = 0 Examples of conservative forces include: Gravity Electromagnetic forces Spring forces Examples of non-conservative forces include: kinetic friction air drag

13 Mechanical Energy When dealing with macroscopic systems (large objects) it is nice to deal with mechanical energy: E mec = K + U We can easily observe changes in E mec (as opposed to changes in internal energy). In order to change E mec for a system, we need to perform some kind of work on the system due to an external force. By lifting a ball, I have changed the potential energy of the system (ball and Earth) and thus the mechanical energy of the system.

14 Mechanical Energy In an isolated system where conservative forces only cause energy changes, E mec is constant for the system. But in this system K and U can change. This is the principle of conservation of mechanical energy. In equation form: If W nc = 0, ΔE mec = ΔK + ΔU = 0 Thus, if the work done on the system is zero, then for two time periods 1 and 2: K 1 + U 1 = K 2 + U 2

15 Mechanical Energy Example A block slides across a horizontal, frictionless floor with an initial velocity of 3.0m/s, it slides up a ramp which makes a 30 o angle to the horizontal floor. What distance will it travel up the ramp? v o 30 o Δy Answer First, you must define a coordinate system. Let s choose the upward direction as positive and make the floor to be y = 0.

16 Mechanical Energy Answer Next, use conservation of mechanical energy: While on the floor: E floor = U 1 + K 1 = 0 + K 1 = (1/2)mv o 2 While at its maximum height: E max = U 2 + K 2 = U = mgδy Since no energy is taken away from the system between the initial and final points:

17 Answer Mechanical Energy Make a triangle to solve for the distance up the ramp: d 30 o Δy = 0.46m

18 Conceptual Question Blocks A and B, of equal mass, start from rest and slide down the two frictionless ramps shown below. Their speeds at the bottom are v A and v B. Which of the following equations regarding their velocities is true? A B 1m 1m 30 o 60 o A) v A > v B B) v A = v B C) v A < v B

19 For Next Time (FNT) Finish the homework for Chapter 6. Start reading Chapter 7.