Taylor 4.2-4.4 (today) and 4.3-4.5 (Wednesday).
For a system of n particles the kinetic energy is the sum of the individual kinetic energies:
We can follow the same argument used to derive the work -- kinetic energy theorem for a single particle to derive it for a system of particles. I will omit the (more complex) derivation here, and refer you instead to the discussion in section 4.10 of Taylor.
Let's look at the kinetic energy for the system of particles. If we express vi in terms of the CM velocities, we can break the kinetic energy into a piece due to the motion of the CM and a piece due to the relative motions of the particles.
Inserting this into the kinetic energy expression, and noting that the remaining dot product (third term) sums to zero, we get:
In words, the total kinetic energy of a system is equal to the kinetic energy due to the motion of the CM plus the kinetic energy due to the motion of particles of the system relative to the CM. This latter part can come from rotation of the system (a collective motion of all the particles of the system), or the motion of individual particles of the system.
If the system of particles are part of a rigid body, then all the relative velocities, v'i, are due to the object's rotation. At this point, we will assume that the rotation is about a fixed axis. about the CM and can be related to the angular velocity and distance from the axis of rotation (passing through the CM), v'i = ωr⊥i:
where the definition of the moment of inertia was used to simplify the result. Note the similarity between the equations for rotational and translational kinetic energy. It is possible to generalize the work -- kinetic energy theorem to include rotational kinetic energy, following steps similar to the earlier derivation.
| relation | translational motion | rotational motion |
|---|---|---|
| momentum | px = mvx | Lz = Izω |
| kinetic energy | T = 1/2 mv² | T = 1/2 Izω² |
| equation of motion | Fx = dpx/dt | Nz = dLz/dt |
Calculate the kinetic energies of CM motion and relative motion for a rod of mass m and length l swinging from one end with angular velocity ω.
The center of mass is in the middle of the rod, a distance of l/2 from the fixed end. The moment of inertia of the rod about the CM is ml²/12. The moment of inertia of the rod about an end is ml²/3. This example has no net translational motion, so that the kinetic energy can be found in two ways: as the sum of the kinetic energy of the motion of the CM plus rotation about the CM; or as the kinetic energy of rotation about the fixed point. The kinetic energy of the motion of the CM is
and the kinetic energy of the rotation about the CM is
The final sum is the moment of inertia of the rod about the CM, which can be taken to an integral for a continuous object
Calculating the kinetic energy as the rotational kinetic energy about the fixed end yields
A quick check confirms that Ttot = TCM + Trel.
You probably recall that potential energy is a useful concept, but that not every force has an associated potential energy. Forces which can be associated with a potential energy are called conservative forces. Gravity, constant or 1/r², and the electrostatic force are conservative, while friction isn't. We'd like a simple way of differentiating conservative forces from non-conservative forces.
First consider a force in one dimension, F = F xhat where F is some scalar function. The idea of a potential energy is to associate a scalar function, U(x) with the force F such that F = -dU(x)/dx. Then, the integral for the work simplifies as
That is, the work done in moving from position 1 to position 2 only depends on the value of a function at points 1 and 2, and nothing in between. Furthermore, for the potential energy to be a useful concept, the function U can't depend on time (we don't want to get a different value for the work if we move the particle now, or a litle bit later) or velocity (we don't want a different value for the work if we move quickly from point 1 to 2 or more slowly). What we want is a function that only depends on position, and determines the total work required to move from point 1 to point 2 independent of the path taken, time of motion, or velocity of the motion. We can easily generalize this to three dimensions as follows:
A force F acting on a particle (object) is conservative if and only if it satisfies two conditions:
A conservative force has an associated potential energy function U(r). We can then define a particle's mechanical energy (or simply energy) as the sum of the particle's kinetic energy and its potential energy:
and as long as there are no non-conservative forces acting on the particle, the total energy is conserved. This is a direct consequence of the work -- kinetic energy theorem:
or after rearranging:
and since 1 and 2 can be any points, this is only true if T + U = E is constant.