به انگلیسی

Trading velocity for momentum

Science · Forces & Motion · ages 22-23

Name ______________________ Date ____________

For L = 2v² − 9x², what is the conjugate momentum p = ∂L/∂v?
- 4v
- 2v
- −9x
A cart obeys 2ẍ = −8x, with L = v² − 4x² and p = 2v. Which canonical pair follows?
- ẋ = 2p and ṗ = −8x
- ẋ = p/2 and ṗ = −8x
- ẋ = p/2 and ṗ = −4x
The Lagrangian is L equals one half m v squared minus one half k x squared, with v equals x prime. The conjugate momentum is p equals dL/dv. What is p?
- m v
- -k x
- m v minus k x
Hamilton's equations hold, and the Hamiltonian H(x, p) carries no explicit time dependence. Its value stays constant along the motion.
Circle one: True False
For the spring oscillator with H = p²/(2m) + ½kx² and m > 0, the state point sits where x > 0 and p > 0. Which way does x move next?
- Leftward, since x is shrinking
- Rightward, since x is growing
- It stays put
For the oscillator with H = p²/2 + x² (m = 1, k = 2), with x horizontal and p vertical, the state sits at x = 1, p = 1. Which way does it move?
- Up and to the left
- Down and to the right
- Straight down, with no horizontal motion
For H = p²/4 + 4x² with m = 2, k = 8, the energy is E = 16. What is the x-extent of the phase ellipse?
- x = ±4
- x = ±8
- x = ±2
For H equals p squared over 2m plus one half k x squared, the rule x prime equals dH/dp gives which result?
- minus k x
- k x
- p over m
Two spring carts with the same m and k are drawn on one phase plot with the same axes. One ellipse sits fully inside the other. What can you say about their energies?
- The inner loop carries the larger energy
- The inner loop has the smaller energy
- Both loops must carry equal energy
A particle has mass 4 kg and velocity 3 m/s, and its conjugate momentum is p = m v. Type the value of p in kg·m/s.
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

Trading velocity for momentum W1-mt_9bMDXQiq_n-s1

4v · Differentiating 2v² with respect to v gives 4v; the −9x² term has no v and contributes nothing.
ẋ = p/2 and ṗ = −8x · p = 2v inverts to v = p/2, and H = p v − L = p²/4 + 4x², so ẋ = ∂H/∂p = p/2 and ṗ = −∂H/∂x = −8x.
m v · Only the velocity term responds to dL/dv, and its derivative is m v.
True · With no stray t inside H, Hamilton's equations give dH/dt = 0, so H is conserved; it equals T + V for ordinary time-independent coordinates with a natural Lagrangian.
Rightward, since x is growing · Positive p makes ẋ = p/m positive, so x grows and the point drifts rightward; the vertical motion is a separate question.
Down and to the right · ẋ = ∂H/∂p = p = 1 > 0 and ṗ = −∂H/∂x = −2x = −2 < 0, so the flow is down and to the right.
x = ±2 · At p = 0 the condition 4x² = 16 gives x² = 4, so x = ±2.
p over m · Differentiate with respect to p: the x term drops out and p squared over 2m gives p over m.
The inner loop has the smaller energy · With the same m, k, and axes, a bigger loop means a larger fixed value E, so the inner loop has the smaller energy.
12 · Multiply mass by velocity: 4 kg times 3 m/s is 12 kg·m/s.