Lecture 10 Potential energy and conservation of energy
Today s Topics: Potential Energy and work done by conservative forces Work done by nonconservative forces Conservation of mechanical energy Potential energy curves and equipotentials
Potential Energy Potential Energy has the potential to do work! A change in Potential Energy arises from the work done by a conservative force. The work done by gravity when a ball is lifted to height Δh does work on the ball when it is released. If the work done by gravity in lifting the ball a height Δh is (-mgδh), ΔPE -W conservative mgδh
Example Δh θ m A box of mass m is pushed to the top of a rough board of length L, that is inclined by θ to the horizontal. What is the change in potential energy of the box? F F N mg Same change in potential as raising the box by Δh f Identify the forces acting on the box Identify the conservative forces that do work F N and mgcosθ are perpendicular to the displacement, so they do no work. F and f are not conservative forces, so they can not change the potential energy W mg - mgsinθ L - mgδh ΔPE - W mg mgδh
ACT: Up the Hill Two paths lead to the top of a big hill. One is steep and direct, while the other is twice as long but less steep. How much more potential energy would you gain if you take the longer path? a) the same b) twice as much c) four times as much d) half as much e) you gain no PE in either case Because your vertical position (height) changes by the same amount in each case, the gain in potential energy is the same.
ACT: Sign of the Energy Is it possible for the gravitational potential energy of an object to a) yes b) no be negative? Gravitational PE is mgh, where height h is measured relative to some arbitrary reference level where PE 0. For example, a book on a table has positive PE if the zero reference level is chosen to be the floor. However, if the ceiling is the zero level, then the book has negative PE on the table. Only differences (or changes) in PE have any physical meaning.
Back to the Work-Energy Theorem In normal situations both conservative and nonconservative forces act simultaneously on an object, so the work done by the net external force can be written as W W c + W nc W KE f - KE o DKE W c -DPE DKE -DPE +W nc THE WORK-ENERGY THEOREM W nc DKE + DPE
Conservation of Mechanical Energy W nc ( KE - KE ) + ( PE PE ) DKE + DPE - f o f o W nc ( KE + PE )- ( KE + PE ) f f o o W nc Ef - E o If the net work on an object by non-conservative forces is zero, then its energy does not change:
The Pendulum A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass). The angle it makes with the vertical varies with time as a sine or cosine. DEMO: Bowling ball pendulum If we displace a pendulum by an angle, q, away from equilibrium, it rises by an amount: Dh L - Lcosq L(1 -cosq ) aboveits initial position. DPE mgdh mgl(1 -cosq )
ACT: Down the Hill Three blocks of equal mass start from rest and slide down different frictionless ramps. All ramps have the same height. Which block has the greatest speed at the bottom of its ramp? d) same speed for all blocks a b c All of the blocks have the same initial gravitational PE, because they are all at the same height (PE mgh). Thus, when they get to the bottom, they all have the same final KE, and hence the same speed (KE mv ). 1
ACT: Water Slide I Paul and Kathleen start from rest at the same time on frictionless water slides with different shapes. At the bottom, whose velocity is greater? a) Paul b) Kathleen c) both the same Conservation of Energy: E i mgh E f mv 1 1 therefore: gh v Because they both start from the same height, they have the same velocity at the bottom.
ACT: Water Slide II Paul and Kathleen start from rest at the same time on frictionless water slides with different shapes (but equal lengths). Who makes it to the bottom first? a) Paul b) Kathleen c) both the same Even though they both have the same final velocity, Kathleen is at a lower height than Paul for most of her ride. Thus, she always has a larger velocity during her ride and therefore arrives earlier!
ACT: Runaway Truck A truck, initially at rest, rolls down a frictionless hill and attains a speed of 0 m/s at the bottom. To achieve a speed of 40 m/s at the bottom, how many times higher must the hill be? a) half the height b) the same height c) times the height d) twice the height e) four times the height Use energy conservation: Ø initial energy: E i PE g mgh Ø final energy: E f KE mv Conservation of Energy: E i mgh E f mv 1 therefore: gh v So if v doubles, H quadruples! 1 1
Let s continue our Example Δh m θ Now our box is at rest at the top of the inclined plane and we release it. First, let s ignore friction. What is it s initial and final KE? What is its initial and final PE? What is its final velocity? KE 0 0 since box is at rest PE 0 mgδh PE f 0 since box is at bottom KE F mgδh since we are ignoring friction and, so, the total mechanical energy is conserved PE 0 + KE 0 mgdh + 0 mgdh v PE 0 + 1 F 1 mv + mv gdh KE F
If we included friction: f m Δh θ Then work is done by friction, and the final velocity is smaller W W NC NC E 1 F - mv E 0 - mgdh
Storing potential energy in a spring: PE spring 1 kx
ACT: Elastic Potential Energy How does the work required to stretch a spring cm compare with the work required to stretch it 1 cm? a) same amount of work b) twice the work c) four times the work d) eight times the work The elastic potential energy is kx. So in the second case, the elastic PE is four times greater than in the first case. Thus, the work required to stretch the spring is also four times greater. 1
ACT: Springs and Gravity A mass attached to a vertical spring causes the spring to stretch and the mass to move downwards. What can you say about the spring s potential energy (PE s ) and the gravitational potential energy (PE g ) of the mass? a) both PE s and PE g decrease b) PE s increases and PE g decreases c) both PE s and PE g increase d) PE s decreases and PE g increases e) PE s increases and PE g is constant The spring is stretched, so its elastic PE increases, because PE s 1 kx. The mass moves down to a lower position, so its gravitational PE decreases, because PE g mgh.
Potential Energy Curves and Equipotentials The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy: DEMO: Wavy track
The potential energy curve for a spring:
Contour maps are also a form of potential energy curve: