Classical Mechanics I (Spring 2019): Homework #1...

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Classical Mechanics I (Spring 2019): Homework #1 Solution Due Mar. 28, 2019 [0.5 pt each, total 6 pts] 1. Thornton & Marion, Problem 2-7 (Note: For Problems 2-7 and 2-33, you will need to use Eq.(2.21) with ρ air =1.23 kg m -3 .) m¨ x = - 1 2 c W ρA ˙ x p ˙ x 2 y 2 , m¨ y = -mg - 1 2 c W ρA ˙ y p ˙ x 2 y 2 The bales should be released 180 m behind the cattle as opposed to 210 m you saw in the in-class quiz, Problem 2-6(a). 2. Thornton & Marion, Problem 2-22 (b) ¨ x = ω c ˙ y, ¨ y = -ω c x - E y /B), ¨ z = qE z /m (c) ... x = -ω 2 c x - E y /B) ˙ x = C 1 cos ω c t + C 2 sin ω c t + E y /B 3. Thornton & Marion, Problem 2-25 (a) normal force N = mg + mv 2 R = mg + 2mgh R (b) N = mg cos 45 + mv 02 R = mg 2 + m R h 2gh - 2gR 1 - 1 2 i 4. Thornton & Marion, Problem 2-33 (a) m¨ y = -mg + 1 2 c W ρA ˙ y 2 (d) The ping-pong ball and the raindrop reach their terminal speeds. But the baseball hits the ground before reaching its terminal speed. 5. Thornton & Marion, Problem 2-44 (Note: For Problems 2-44, derive step by step the equation of motion depicting the small oscillation; this problem was briefly discussed in the class.) Please see the scanned image attached. 1

Transcript of Classical Mechanics I (Spring 2019): Homework #1...

Classical Mechanics I (Spring 2019): Homework #1 Solution

Due Mar. 28, 2019

[0.5 pt each, total 6 pts]

1. Thornton & Marion, Problem 2-7

(Note: For Problems 2-7 and 2-33, you will need to use Eq.(2.21) with ρair = 1.23 kg m−3.)

• mx = −12cWρAx

√x2 + y2, my = −mg − 1

2cWρAy√x2 + y2

• The bales should be released ∼180 m behind the cattle as opposed to ∼210 m you saw in thein-class quiz, Problem 2-6(a).

2. Thornton & Marion, Problem 2-22

• (b) x = ωcy, y = −ωc(x− Ey/B), z = qEz/m

• (c)...x = −ω2

c (x− Ey/B) → x = C1 cosωct+ C2 sinωct+ Ey/B

3. Thornton & Marion, Problem 2-25

• (a) normal force N = mg + mv2

R = mg + 2mghR

• (b) N = mg cos 45◦ + mv′2

R = mg√2

+ mR

[2gh− 2gR

(1− 1√

2

)]

4. Thornton & Marion, Problem 2-33

• (a) my = −mg + 12cWρAy

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• (d) The ping-pong ball and the raindrop reach their terminal speeds. But the baseball hitsthe ground before reaching its terminal speed.

5. Thornton & Marion, Problem 2-44

(Note: For Problems 2-44, derive step by step the equation of motion depicting the smalloscillation; this problem was briefly discussed in the class.)

• Please see the scanned image attached.

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6. Thornton & Marion, Problem 3-21

• x = x0 + (v0 + βxo)te−βt, v = [v0 − β(v0 + βxo)t]e

−βt

7. Thornton & Marion, Problem 3-24

• The amplitude of oscillation, Eq.(3.59), of the particular solution xp(t) peaks at ωR, Eq.(3.63).

8. Thornton & Marion, Problem 3-26

• m1x1 + b1(x1 − x2) + kx1 = F cosωt, m2x2 + b1(x2 − x1) + b2x2 = 0

• L1q1 +R1(q1− q2) + (1/C)q1 = E0 cosωt, L2q2 +R1(q2− q1) +R2q2 = 0 if the substitutionsseen in Examples 3.4 and 3.5 are applied.

9. In the class we discussed two-dimensional simple harmonic oscillators. (a) Describe anexample of the physical realization of Eqs. (3.19) and (3.27). (b) Reproduce Figures 3-2 to 3-4in your textbook with your favorite numerical tool. For Figure 3-2, you don’t have to remakeall 10 panels; 2-3 panels should be enough.

• A mass m is tied to four springs, two of constant k1 in the x-direction, and two others ofconstant k2 in the y-direction. One end of each of the springs is tied to m (in a + shape), andthe other end is tied to walls. At equilibrium none of the springs is stretched. We consider onlysmall oscillations from the equilibrium point.

10. Show that the logarithmic decrement of the oscillations in an RLC circuit is approximatelyπR√C/L if the resistance is small.

• βτ1 = (b/2m)(2π/ω1) ' (b/2m)(2π/ω0) = πb√

1/mk for a lightly damped mechanical oscil-lator. Then the correspondence between mechanical and electrical oscillators is applied.

11. Consider a particle of mass m under the influence of the potential U(x) = U0(−ax2 + bx4)where U0 = 1 J, and a and b are positive constants. (a) Plot both U(x) and the force acting onthe particle, F (x), numerically. (b) Find the equilibrium points. (c) Determine their stabilities.(d) If a stable equilibrium point exists, determine the period of small oscillations if the particleis given a small displacement from the equilibrium point.

• (c) At x = x0 = ±√

a2b , [d2U/dx2]x=x0 = 4aU0 > 0

• (d) T = 2π√

m4aU0

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12. Consider the system of a pulley and two masses illustrated below. A massless string of lengthb is attached to a block of mass m1, runs over a massless, frictionless pulley, then attached to ametal ball of mass m2. The ball, with a hole through it, is threaded on a frictionless vertical rod.The rod and the pulley are separated by d. Assume that the sizes of the pulley and the ball arenegligible. (a) Using the variable θ shown below, find U(θ). (b) Find the equilibrium point(s).What condition should m1 and m2 meet for the equilibrium to occur? (c) If equilibrium pointsdo exist, determine their stabilities.

• (a) U(θ) = m1gd/sin θ −m2gd/tan θ + C

• (b) For θ0 = cos−1 (m2/m1) to be realistic, m1 > m2.

• (c) [d2U/dθ2]θ=θ0 =[(m1sin

2θ − 2 cos θ(m2 −m1cos θ)) · gdsin3θ

]θ=θ0

= m1gdsin θ0

> 0

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