Force
If we are given a potential energy, how do we find the force? (for a conservative force)
The force is the gradient of the potential energy function.
When can a potential energy function be approximated as a simple harmonic oscillator potential?
Near a local minimum in the function.
Write down the formula for the energy, force, and the position function of a simple harmonic oscillator.
U = 0.5kx2
F = -kx
x(t) = Acos(wt + δ)
Find, in your notes somewhere, the formula for the A and δ in terms of Fo, m, wo, w, and β.
A = Fo/m*sqrt((wo2-w2)2+4β2w2)
δ = arctan(2βw/(wo2-w2))
Sketch a graph giving an example of underdamped, overdamped, and critically damped oscillation.
(graph).
Calculate the gradient of the following functions, f(x, y, z): (a) f = ln(r), (b) f = rn, (c) f = g(r), where r = sqrt(x2+y2+z2) and g(r) is some unspecified function of r. [Hint: Use the chain rule.] (Taylor 4.13)
(a) r-hat / r
(b) nrn-1 r-hat
(c) g'(r) r-hat
A mass m moves in a circular orbit (centered on the origin) in the field of an attractive central force with potential energy U = krn. Prove the virial theorem that T = nU/2. (Taylor 4.41)
Since the mass moves in a circle, the radial component of its accelerations is just —v2/r, the centripetal acceleration. By Newton’s second law, m(—v2/r) = Fr = —dU/dr = —nkrn-1, from which we see that mv2 = nkrn = nU, and hence T = 0.5mv2 = 0.5nU.
(a) If a mass m = 0.2 kg is tied to one end of a spring whose force constant k = 80 N/m and whose other end is held fixed, what are the angular frequency ω, the frequency f, and the period τ of its oscillations? (b) If the initial position and velocity are x0= 0 and v0= 40 m/s, what are the constants A and δ in the expression x(t) = A cos(ωt-δ)? (Taylor 5.8)
(a) ω=20 s-1, f=3.2 Hz, τ=0.31s
(b) δ=π/2, A=2 m
Write down four equivalent ways to represent simple harmonic motion in one dimension.
x(t) = C1eiwt + C2e-iwt
x(t) = B1cos(wt) + B2sin(wt)
x(t) = Acos(wt - δ)
x(t) = Re(Ceiwt)
Find the partial derivatives with respect to x, y, and z of the following functions: (a) f (x, y, z) = ax2 + bxy + cy2, (b) g(x, y, z) = sin (axyz2), (c) h(x, y, z) = aexy/z^2, where a, b, and c are constants. (Taylor 4.10)
(too much for me to copy down from the solution manual lol)
If a particle's potential energy is U(r) = k(x2 + y2 + z2), where k is a constant, what is the force on the particle? (Taylor 4.16)
F=-2k(x, y, z)=-2kr
A mass moves in an arbitrary path from point 1 (height 0) to point 2 (height h) in a uniform gravitational field g. Find the work done by gravity on the mass, and find a general formula for the gravitational potential energy in the uniform gravitational field. (Taylor 4.5)
Work: W=-mgh
Potential energy: U=mgy
Write down the potential energy U(φ) of a simple pendulum (mass m, length l) in terms of the angle φ between the pendulum and the vertical. (Choose the zero of U at the bottom.) Show that, for small angles, U has the Hooke's law form U(φ) = 0.5 φ2, in terms of the coordinate φ. What is k? (Taylor 5.3)
The height of the mass below the pivot is lcosφ. Therefore the height above the bottom is l(1 — cosφ) and the PE is U = mgl(1 — cosφ). If φ is small, cosφ =~ 1 - 0.5φ2 and U ~= 0.5mglφ2 = 0.5kφ2, where k = mgl.
A damped oscillator satisfies the equation (ma+bv+kx=0), where Fdmp= —bv is the damping force. Find the rate of change of the energy E = 0.5mv2+0.5kx2 (by straightforward differentiation), and, with the help of (ma+bv+kx=0), show that dE/dt is (minus) the rate at which energy is dissipated by Fdmp. (Taylor 5.23)
Prove that if f (r) and g(r) are any two scalar functions of r, then ∇(fg) = f∇g + g∇f. (Taylor 4.14)
(do the proof)
Which of the following forces is conservative? (a) F = k(x , 2y, 3z) where k is a constant. (b) F = k(y, x, 0). (c) F = k(—y, x, 0). For those which are conservative, find the corresponding potential energy U, and verify by direct differentiation that F = — ∇U. (Taylor 4.23)
(a) Conservative: U=-k(0.5x2+y2+1.5z2)
(b) Conservative: U=-kxy
(c) Not conservative.
EVIL PROBLEM ALERT!!! (Taylor 4.8)
Consider a small frictionless puck perched at the top of a fixed sphere of radius R. If the puck is given a tiny nudge so that it begins to slide down, at what angle from the top of the sphere will it begin to part from the surface of the sphere? [Hint: Use conservation of energy to find the puck's speed as a function of its angle, then use Newton's second law to find the normal force of the sphere on the puck. At what value of this normal force does the puck leave the sphere?]
θ=arccos(2/3)=48.2°
Consider a particle in two dimensions, subject to a restoring force Fx=-kx, Fy=-ky. Prove that its potential energy is U=0.5(kxx2+kyy2). (Taylor 5.14)
(Integrate the force in x and y to find the potential energy)
The solution for x(t) for a driven, underdamped oscillator is most conveniently found in the form x(t) = A cos(wt — δ) + e-βt[B1cos(w1t) + B2sin(w2t)] . Solve that equation and the corresponding expression for v, to give the coefficients B1 and B2 in terms of A, δ, and the initial position and velocity x0 and v0. (Taylor 5.33)
B1 = x0 - Acosδ
B2 = (1/w1)(v0 - Awsinδ + βB1)
(a) Prove that any complex number z = x + iy (with x and y real) can be written as z = reiθ where r and θ are the polar coordinates of z in the complex plane. (Remember Euler's formula.) (b) Prove that the absolute value of z, defined as |z| = r, is also given by |z|2= zz*, where z* denotes the complex conjugate of z, defined as z* = x — iy. (c) Prove that z* = re-iθ (d) Prove that (zw)* = z*w* and that (1/z)* = 1/z*. (Taylor 5.35)