Tutorial 5 PDF

Preview

Summary

tutorial...

Description

Math 235, Tutorial #5 23 March 2018

1. Evaluate the work done W =

Z

P

O 2

~F · d~r =

Z

P

(Fx dx + Fy dy) O

by the two-dimensional force ~F = (x , 2xy) along the three paths joining the origin O to the point P = (1, 1) as shown in Figure 1 and defined as follows: (a) This path goes along the x axis to Q = (1, 0), and then straight up to P . (Divide the integral into RQ RP RP two pieces, O = O + Q .) (b) On this path y = x2 , and you can therefore replace dy by 2xdx and perform the integral over x. (c) This path is given parametrically as x = t3 , y = t2 . In this case, rewrite x, y, dx, and dy in terms of t and dt, and convert the integral into an integral over t. 2. Consider a mass m in a uniform gravitational field ~g, so that the force on m is m~ g, where ~g is a constant vector pointing vertically down. If the mass moves by an arbitrary path from point 1 to point 2, show that the work done by gravity is Wgrav (1 → 2) = −mgh, where h is the vertical height gained between points 1 and 2. Use this result to prove that the force of gravity is conservative. 3. 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, through what vertical height will it decend before it leaves the surface of the sphere? [Hint: Use conservation of energy to find the puck’s speed as a function of its height, 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?] ~ of the following functions, f (x, y, z). (Remember that to evaluate a partial 4. Calculate the gradient ∇f derivative, ∂f /∂x, you differentiate with respect to the variable in the denominator while treating the other variables as constants.) (a) f = x2 + z 3 (b) f = ky, where k is a constant p (c) f = r = x2 + y2 + z 2 [Hint: Use the chain rule.] (d) f = 1/r

Figure 1: Line integral paths for Problem 3.

5. Which of the following forces are conservative? For those which are conservative, find the corre~ . (For this problem, ~ = −∇U sponding potential energy U, and verify by direct differentiation that F choose the potential energy reference point ~ro to be the origin, (0, 0, 0).) ~ = k(x, 2y, 3z), where k is a constant (a) F ~ = k(y, x, 0) (b) F ~ (c) F = k(−y, x, 0) 6. Consider a simple pendulum, consisting of a point mass m fixed to the end of a massless rod (length l), whose other end is pivoted from the ceiling to let it swing freely in a vertical plane. The pendulum can be treated as a one-dimensional system, and its position can be specified by its angle φ from the equilibrium position. (a) Show that the pendulum’s potential energy (measured from the equilibrium level) is U (φ) = mgl(1 − cos φ). ˙ Show that by differentiating your (b) Write down the total energy E as a function of φ and φ. expression for E with respect to t you can get the equation of motion for φ and that the equation of motion is just the familiar Γ = Iα (where Γ is the torque, I is the moment of inertia, and α is the angular acceleration ¨φ). (c) Assuming that the angle φ remains small p throughout the motion, solve for φ(t) and show that the motion is periodic with period τ = 2π l/g.

7. (a) Verify the following equations that give x, y, z in terms of the spherical polar coordinates r, θ, φ: x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. (b) Find expressions for r, θ, φ in terms of x, y, z .

8. We showed in lecture that a force that is central and conservative is automatically spherically symmetric. In this problem, you will prove the reverse: that is, a force that is central and spherically symmetric is automatically conservative. If a force is central and spherically symmetric, then it must ~ r) = f (r)ˆr. Using Cartesian coordinates, show that this implies that ∇ ~ × ~F = 0. have the form F(~ 9. The potential energy of two atoms in a molecule can sometimes be approximated by the Morse function, U (r) = A[(e(R−r)/S − 1)2 − 1] where r is the distance between the two atoms and A, R, and S are positive constants with S ≪ R. Sketch this function for 0 < r < ∞. Find the equilibrium separation ro at which U (r) is minimum. Now write r = ro + x so that x is the displacement from equilibrium, and show that, for small displacements, U has the approximate form U = const + kx 2 /2. That is, Hooke’s law applies. What is the force constant k ? 10. We have discussed four equivalent ways to represent simple harmonic motion in one dimension: x(t) = C1 eiωt + C2 e−iωt

(1)

= B1 cos(ωt) + B2 sin(ωt)

(2)

= A cos(ωt − δ) = Re[Ceiωt ]

(3) (4)

To make sure you understand all of these, show that they are equivalent by proving the following implications: (1) → (2) → (3) → (4) → (1). For each form, give an expression for the constants (C1 , C2 , etc.) in terms of the constants of the previous form. 11. You are told that, at the known positions x1 and x2 , an oscillating mass m has speeds v1 and v2 . What are the amplitude and angular frequency of the oscillations?...