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Classical Mechanics: Part II 2014

Contents 1 Energy and Potentials

1.1 Potential Energy . . . . . . . . . . . . . . . .

1.2 Conservation of Energy . . . . . . . . . . . .

1.3 The Work-Energy Theorem . . . . . . . . . .

1.4 Motion in a potential . . . . . . . . . . . . . .

1.5 Equilibrium and why (almost) everything is a

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . . Harmonic Oscillator

2 Conservation Laws

2.1 Conservation of Momentum . . . . . . . . . . . .

2.2 Centre of Mass . . . . . . . . . . . . . . . . . . .

2.3 Kinetic Energy and Centre of Mass Motion . . .

2.4 Collisions: Part I . . . . . . . . . . . . . . . . . .

2.5 Conservation of Angular Momentum . . . . . . .

2.6 Angular Momentum and Centre of Mass Motion

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1

ENERGY AND POTENTIALS

1 2

Energy and Potentials

1.1 Potential Energy

Let us consider a force F that depends only on x and not on x, such force is called conservative. For this type of force it's possible to define the Potential Energy V . If we limit ourselves to one-dimension the force is obtained from the potential energy by dV F (x) = -

. (1) dx Note the negative sign in this definition! The relationship can be inverted by integrating both sides obtaining Z x F (x0 )dx0 . (2) V (x) = -

x0

where x0 is an arbitrary reference point (the position of zero potential) needed to fix the integration constant, and x0 is just an integration dummy variable. We have already met a number of force laws, and we can practice deriving the corresponding potentials.

* Hooke's Law:

x Z

k x0 dx0

F = -kx - V =

(3)

0 V =

1 2 kx 2

(4)

This is generally referred to as elastic potential energy, and is the from that potential energy takes when the system exhibits simple harmonic motion.

* Gravity at the Earth's surface: z

Z

m g dz 0

F = -mg - V =

(5)

z 0 =0

V = mgz

(6)

where z is the height measured from the point chosen as the zero of the potential.

* Gravity: F = -G

M1 M2

-V =

x2 V = -G

Z

x G

[?]

M1 M2 x

M1 M2 dx x2

(7)

(8)

Note how by definition the gravitational potential energy comes out to be negative and the point of zero potential is conventionally chosen to be infinity.

1

ENERGY AND POTENTIALS

1.2 3

Conservation of Energy

The utility of the potential formulation will become clear once we define another fundamental quantity in Mechanics, the Kinetic Energy T T =

1 mx2 . 2

(9)

The total energy E of a Mechanical system is E=

1 mx2 + V 2

(10)

We can now prove one of the most famous statements in Mechanics (or at least one version thereof): The total energy E = T + V of a single particle is conserved. To prove it let us consider the rate of change of total energy E over time

dV dV x = x mx +

=0 E = mxx +

dx dx

(11)

Since E does nor change with time, it must be a constant over time, i.e. it's conserved. Conserved quantities are very precious to us, and we will spend quite a lot of time from now on highlighting their different usages. For the moment, if you feel the need to better visualise conservation of energy, try and calculate T and V for the simple harmonic oscillator and show that E is conserved.

Figure 1: A sketch T and V for simple harmonic motion.

1.3 The Work-Energy Theorem

There is an alternative and equally quite useful approach to energy conservation that applies even to non-conservative forces. Let us define the work W (always in one dimension, for the time being) done by a force F acting on a particle by Z W [?] F dx (12)

1

ENERGY AND POTENTIALS

4 Let us change integration variables and use N2 F = mx to get Z Z Z t2 dx d 1 W = F dt (x2 ) = T (t2 ) - T (t1 ) dt = m xxdt = m dt 2 dt t1

(13)

So the change in kinetic energy is equal to the total work done. Since friction is not a conservative force, we can still say that energy is conserved if we take into account the work done by friction, T + V + Wf = const. Note that the potential energy of a body subject to a force field can be defined as the amount of work done by an external force against the force field. Indeed if we compare 12 and 2 they differ only by a negative sign, because the force considered in 12 is the external force applied to the body, while the force in 2 is the force field, which by Newton's third law is equal and opposite. As an example, if we consider a spaceship close to Earth, the force field acting on it is the gravitational attraction of the Earth while the external force would need to be applied by the spaceship engine to counteract gravity. In light of this, it makes sense that the gravitational potential energy is negative, because work is done ON the spaceship and not BY it on approaching the Earth1 .

1.4 Motion in a potential

Let us consider motion in one dimension in an arbitrary potential V (x). Although the equation of motion is a second order differential equation, the existence of a conserved quantity magically turns the problem into a first order differential equation. r dx 2 1 2

=+-

(E - V (x)) (14) E = mx + V (x) -

2 dt m The problem can be solved, at least formally, by integrating both sides and finding x. Naturally, by having turned a second order problem into a first order problem, we must have chosen an integration constant. In this case it is the total energy E. In general, even if we cannot integrate 14, sketching the form of the potential we can tell a lot about the character of the motion. Firstly, since the kinetic energy is always positive, the motion is restricted to values of x for which V (x) < E. Making an gravitational example, if you slide a ball down a hill of height h, it will not make it up a subsequent hill higher than h.2

1.5 Equilibrium and why (almost) everything is a Harmonic Oscillator

In equilibrium a particle feels not net force F = 0; we must have dV

= 0. dx

(15)

1 This is a very important point and it might be worth rereading it in due time. If you are doing any Thermodynamics this year, perhaps in your Chemistry course, exactly the same sign convention applies. 2 Note that this requirement is relaxed in Quantum Mechanics, as we shall see later this year! A quantum particle can tunnel through a region where V (x) > E with some probability. As if our ball, giving up going up the second higher hill, would just dig itself a tunnel under it!

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