Trading velocity for momentum

Science · Forces & Motion · ages 22-23

Name ______________________ Date ____________

A particle has the Lagrangian L = (1/2) m v^2 - (1/2) k x^2, where v = x'. The momentum conjugate to x is p = dL/dv (the partial derivative of L with respect to v). What is p?
- m v
- -k x
- (1/2) m v^2
- m v - k x
A cart on a spring obeys m x'' = -k x. Hamilton's approach trades the velocity for the momentum p = m x'. Which pair of first-order equations replaces the single second-order equation?
- x' = p/m and p' = -k x
- x' = -k x and p' = p/m
- p' = x/m and x' = -k p
- x' = p m and p' = k x
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
If the Hamiltonian H of x and p carries no explicit time dependence, its value stays constant along the motion.
Circle one: True False
The state point sits where x is positive and p is positive. Which way does it move next?
- Leftward, because x is shrinking
- Downward, because p is shrinking
- Rightward, because x is growing
A harmonic oscillator moves around its ellipse in phase space. At a point where x is positive and p is positive, which way is the state point moving?
- To the right, because x is increasing
- To the left, because x is decreasing
- Straight up, because p is increasing
- Straight down, because p is decreasing
A harmonic oscillator has the Hamiltonian H = p^2/(2m) + (1/2) k x^2, and H keeps a constant value E during the motion. In phase space (x on the horizontal axis, p on the vertical axis), what shape does the trajectory trace out?
- An ellipse
- A spiral winding inward
- A straight line through the origin
- A parabola
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 are drawn on one phase plot. One ellipse sits fully inside the other. What can you say about their energies?
- The inner loop carries the larger energy
- The inner loop carries the smaller energy
- Both loops must carry equal energy
A particle has mass 4 and velocity 3, and its conjugate momentum is p equals m times v. Type the value of p.
Answer: ______________

Answer key

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

Trading velocity for momentum W1-mt_9bMDXQiq_n-s1

m v · Only the kinetic term (1/2) m v^2 contains v, and its derivative with respect to v is m v. The potential term depends only on x, so it contributes nothing.
x' = p/m and p' = -k x · Momentum is defined as p = m x', so x' = p/m. Newton's law then says the momentum changes at the rate p' = m x'' = -k x. One second-order equation has become two first-order ones.
m v · Only the velocity term responds to dL/dv, and its derivative is m v.
True · With no stray t inside H, the value is conserved and equals the energy.
Rightward, because x is growing · Positive p means x prime is positive, so x grows and the point drifts rightward.
To the right, because x is increasing · Hamilton's first equation says x' = p/m. With p positive, x' is positive, so x grows and the point slides to the right along the ellipse.
An ellipse · Setting H = E gives p^2/(2mE) + k x^2/(2E) = 1, which is the equation of an ellipse. Energy conservation locks the motion onto that closed curve.
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 carries the smaller energy · A bigger loop means a larger fixed value E, so the inner loop has the smaller energy.
12 · Multiply mass by velocity: 4 times 3 is 12.