18 worked answers · short steps first, explanations when you need them
Chapter 13 Exercises
Textbook pages 274–277
Question 1
Which examples are periodic: (a) one return trip by a swimmer, (b) a displaced bar magnet released, (c) a hydrogen molecule rotating about its centre of mass, (d) an arrow released from a bow?
Steps only
Only a released magnet oscillating and a rotating molecule repeat regularly.
Answer
(b) and (c)
Question 2
Classify each as nearly SHM or periodic but not SHM: (a) Earth's rotation, (b) a mercury column in a U-tube, (c) a ball bearing near the bottom of a smooth bowl, (d) general vibrations of a polyatomic molecule.
Steps only
SHM: (b), (c) for small motion. Earth’s rotation is periodic but not SHM; molecular vibrations are generally more complex.
Answer
(a) periodic, (b) SHM, (c) SHM, (d) generally not SHM
Question 3
From the four x–t plots in Fig. 13.18, identify periodic plots and state their periods.
Steps only
Look for the smallest time shift that repeats the whole graph: plots (b) and (d) repeat after 2 s.
Answer
(b), (d): T=2 s
Question 4
For each function, say whether it is SHM, periodic but not SHM, or non-periodic; give the period if periodic: (a) sinωt−cosωt, (b) sin³ωt, (c) 3cos(π/4−2ωt), (d) cosωt+cos3ωt+cos5ωt, (e) exp(−ω²t²), (f) 1+ωt+ω²t².
Steps only
(a) and (c) are single sinusoids (SHM); (b), (d) repeat but are not SHM; (e), (f) do not repeat.
Answer
T: (a) 2π/ω; (b) 2π/ω; (c) π/ω; (d) 2π/ω.
Question 5
A particle oscillates between A and B, 10 cm apart, with A-to-B positive. Give the signs of velocity, acceleration and force: (a) at A, (b) at B, (c) at midpoint moving toward A, (d) 2 cm from B moving toward A, (e) 3 cm from A moving toward B, (f) 4 cm from B moving toward A.
Steps only
Take the midpoint as x=0: v follows travel direction; a and F have sign opposite to x.
SHM requires a=−ω²x; only (c) has the required negative linear form.
Answer
(c) only
Question 7
For x=Acos(ωt+φ), the initial position is 1 cm and initial velocity is ω cm/s. Find amplitude and phase for cosine and sine forms, where ω=π s⁻¹.
Steps only
Acosφ=1; −A sinφ=1 ⇒ A=√2, φ=−π/4. For sine form, B sinα=1 and B cosα=1 ⇒ B=√2, α=π/4.
Answer
A=B=√2 cm; φ=−π/4; α=π/4
Question 8
A spring balance reads up to 50 kg over a 20 cm scale. A suspended body oscillates with period 0.6 s. Find its weight.
Steps only
k=50×9.8/0.20=2450 N/m; m=k(0.6/2π)²≈22.3 kg; W=mg≈219 N
Answer
About 219 N
Question 9
A 3 kg mass on a spring of constant 1200 N/m is pulled 2.0 cm and released. Find (i) frequency, (ii) maximum acceleration, (iii) maximum speed.
Steps only
ω=√(k/m)=20 rad/s; f=ω/2π=3.18 Hz; amax=ω²A=8.0 m/s²; vmax=ωA=0.40 m/s
Answer
3.18 Hz; 8.0 m/s²; 0.40 m/s
Question 10
For the spring in Exercise 13.9, take the unstretched position as x=0 and right as positive. Write x(t) if at t=0 the mass is (a) at mean position, (b) maximally stretched, (c) maximally compressed. Compare the functions.
Steps only
A=0.020 m and ω=20 rad/s; select sine/cosine sign to match the starting position and direction.
For the two circular motions in Fig. 13.20, find the corresponding x-projection SHM from each circle's radius, period, initial position and rotation direction.
For each SHM, describe the reference circle (radius, initial point and angular speed), taking anticlockwise rotation: (a) x=−2sin(3t+π/3), (b) x=cos(π/6−t), (c) x=3sin(2πt+π/4), (d) x=2cosπt.
Steps only
Convert each expression to Rcos(ωt+θ₀); R is circle radius and θ₀ is the initial radius-vector angle.
Answer
(a) (2,5π/6,3); (b) (1,−π/6,1); (c) (3,−π/4,2π); (d) (2,0,π), in cm, radians, rad/s.
Question 13
Compare (a) a spring of constant k fixed at one end with mass m at the other, and (b) the same spring with identical masses m attached at both free ends. Find maximum extension under force F and the period after release.
Steps only
The same force F is the spring tension in both arrangements. In (b), relative motion makes the restoring constant 2k for each mass.
Answer
Extension F/k in each; Tₐ=2π√(m/k), Tᵦ=2π√(m/2k).
Question 14
A locomotive piston has stroke 1.0 m and angular frequency 200 rad/min. Find its maximum speed if it moves in SHM.
Steps only
vmax=ωA=(200/60)(1.0/2)=1.67 m/s
Answer
1.67 m/s
Question 15
A pendulum has a 3.5 s period on Earth. Find its period on the Moon where g=1.7 m/s²; on Earth g=9.8 m/s².
Steps only
Tᴹ=3.5√(9.8/1.7)≈8.4 s
Answer
8.4 s
Question 16
A pendulum of length l is inside a car moving at speed v on a circular track of radius R. Find its small-oscillation period when it oscillates radially about equilibrium.
Steps only
gₑff=√(g²+v⁴/R²); T=2π√(l/gₑff)
Answer
2π√[l/√(g²+v⁴/R²)]
Question 17
Show that a floating cylindrical cork of density ρ, area A and height h in liquid of density ρₗ oscillates vertically in SHM when pushed down slightly. Find its period.
Steps only
F=−ρₗgAx; m=ρAh; a=−[ρₗg/(ρh)]x ⇒ T=2π√(ρh/ρₗg)
Answer
2π√(ρh/(ρₗg))
Question 18
Show that after a small pressure difference is removed, mercury in a U-tube oscillates in SHM.