physics 1d03 - lecture 22 potential energy serway and jewett 8.1 – 8.3 work and potential energy...
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Physics 1D03 - Lecture 22
Potential Energy
Serway and Jewett 8.1 – 8.3
• Work and potential energy• Conservative and non-conservative forces• Gravitational and elastic potential energy• Conservation of Mechanical Energy
Physics 1D03 - Lecture 22
Gravitational Work
s2 y s1
mg When the block is lowered, gravity does work:
Wg1 = mg.s1 = mgy
or, taking a different route:
Wg2 = mg.s2 = mgy
y
mg
FP = mg
To lift the block to a height y requires work (by FP :)
WP = FPy = mgy
Physics 1D03 - Lecture 22
Work done (against gravity) to lift the box is “stored” as
gravitational potential energy Ug:
Ug =(weight) x (height) = mgy (uniform g)
When the block moves,
(work by gravity) = P.E. lost
Wg = -Ug
• The position where Ug = 0 is arbitrary.
• Ug is a function of position only. (It depends only on the relative positions of the earth and the block.)
• The work Wg depends only on the initial and final heights, NOT on the path.
Physics 1D03 - Lecture 22
Conservative Forces
A force is called “conservative” if the work done (in going from some point A to B) is the same for all paths from A to B.
An equivalent definition:
For a conservative force, the work done on any closed path is zero. Later you’ll see this written as:
Total work is zero.
path 1
path 2A
B
W1 = W2
0sdF
Physics 1D03 - Lecture 22
Concept Quiz
a) Yes.b) No.c) We can’t really tell.d) Maybe, maybe not.
The diagram at right shows a force which varies with position. Is this a conservative force?
Physics 1D03 - Lecture 22
For every conservative force, we can define a potential energy function U so that
WAB U UA UB
Examples:
Gravity (uniform g) : Ug = mgy, where y is height
Gravity (exact, for two particles, a distance r apart): Ug GMm/r, where M and m are the masses
Ideal spring: Us = ½ kx2, where x is the stretch
Electrostatic forces (we’ll do this in January)
Note the negative
Physics 1D03 - Lecture 22
Non-conservative forces:• friction• drag forces in fluids (e.g., air resistance)
Friction forces are always opposite to v (the directionof f changes as v changes). Work done to overcome friction is not stored as potential energy, but converted to thermal energy.
Physics 1D03 - Lecture 22
If only conservative forces do work, potential energy is converted into kinetic energy or vice versa, leaving the total constant. Define the mechanical energy E as the sum of kinetic and potential energy:
E K + U = K + Ug + Us + ...
Conservative forces only: W UWork-energy theorem: W KSo, KU 0; which means
E is constant in time (ie: dE/dt=0)
Conservation of mechanical energy
Physics 1D03 - Lecture 22
Example: Pendulum
vf
The pendulum is released from rest with the string horizontal.
a) Find the speed at the lowest point (in terms of the length L of the string).
b) Find the tension in the string at the lowest point, in terms of the weight mg of the ball.
L
Physics 1D03 - Lecture 22
Example: Pendulum
vf
The pendulum is released from rest at an angle θ to the vertical.
a) Find the speed at the lowest point (in terms of the length L of the string).
θ
Physics 1D03 - Lecture 22
Example: Block and spring.v0
A block of mass m = 2.0 kg slides at speed v0 = 3.0 m/s along a frictionless table towards a spring of stiffness k = 450 N/m. How far will the spring compress before the block stops?
Physics 1D03 - Lecture 22
Example
From where (what angle or height) should the pendulum be released from rest, so that the string hits the peg (located at L/2) and stops with the string horizontal?
L/2
L/2
Physics 1D03 - Lecture 22
Example
m1m2
Two masses are connected over apulley of Radius R and moment of inertia I. The system is released from rest and m1 falls througha distance of h.
Find the linear speeds of the masses.
Physics 1D03 - Lecture 22
Solution
Physics 1D03 - Lecture 22
Summary
• Conservative and non-conservative forces• Potential energy : work = P.E. lost• Gravitational and elastic (spring) P.E.• Mechanical energy in conservative systems.