# 5月7日 考前猜題一
###### tags: `普通物理`
## Chapter 15. Oscillations
### Problem 1 (§15-2 – §15-4)

A block whose mass $m$ is $540~\text{g}$ is fastened to a spring whose spring constant $k$ is $72~\text{N/m}$. The block is pulled a distance $x =12~\text{cm}$ from its equilibrium position at $x=0$ on a frictionless surface and released from rest at $t=0$.
1. What are the angular frequency, the frequency, and the period of the resulting motion?
2. What is the amplitude of the oscillation?
3. What is the maximum speed $v_m$ of the oscillating block, and where is the block when it has this speed?
4. What is the magnitude $a_m$ of the maximum acceleration of the block?
5. What is the phase constant $\phi$ for the motion?
6. What is the displacement function $x(t)$ for the spring-block system?
7. What is the total mechanical energy $E$ of the spring-block system?
8. What is the block’s speed as it passes through the equilibrium point?

### Problem 2 (§15-2 – §15-4)
At $t=0$, the displacement $x(0)$ of the block in a spring-block system linear oscillator is $25.00~\text{cm}$. The block’s velocity $v(0)$ then is $-0.400~\text{m/s}$, and its acceleration $a(0)$ is $+60.0~\text{m/s}^2$.
1. What is the angular frequency $\omega$ and the period $T$ of this system?
2. What is the phase constant $\phi$?
3. What is the amplitude $x_m$?
4. What is the displacement of the block at $t=2.70 T$?
5. What is the displacement function $x(t)$ for the spring-block system?
6. If the block's mass is $40.0~\text{g}$, then what is the total mechanical energy $E$ of system?
7. What is the kinetic energy of the block as it is at $x=0.60x_m$?
### Problem 3 (§15-5 – §15-6)

In Fig. 15-11a, a stick of length $L=4.00~\text{m}$ swings about a pivot point at one end, at distance $h$ from the stick’s center of mass.
1. What is the period of oscillation $T$?
2. What is the distance $L_0$ between the pivot point $O$ of the stick and the *center of oscillation* of the stick?

### Problem 4 (§15-8)
For the damped oscillator of Fig. 15-14, $m=300~\text{g}$, $k=60~\text{N/m}$, and $b=30~\text{g/s}$.
1. What is the period of the motion if the oscillator is undamped?
2. How long does it take for the amplitude of the damped oscillations to drop to half its initial value?
3. How long does it take for the mechanical energy to drop to one-half its initial value?

## Chapter 16: Waves I
### Problem 1 (§16-2 – §16-8)
A wave traveling along a stretched ideal string is described by the wave function $$y(x,t)=0.00250\sin(20.0x-1.50t)\tag{1}$$ in which the numerical constants are in SI units ($0.00250~\text{m}$, $20.0~\text{rad/m}$, $1.50~\text{rad/m}$). Answer the questions 1–4.
1. What is the amplitude of this wave?
> $y_m=\boxed{0.00250~~\text{m}}$
2. What is the angular wave number, angular frequency and phase constant of this wave?
> - $k=\boxed{20.0~\text{rad/m}}$,
> - $\omega=\boxed{1.50~\text{rad/m}}$,
> - $\phi=\boxed{0~\text{rad}}$.
3. What are the wavelength, period, and frequency of this wave?
> - $\lambda=\dfrac{2\pi}{k}=\dfrac{2\pi~\text{rad}}{20.0~\text{rad/m}}=\boxed{0.314~\text{m}}$,
> - $T=\dfrac{2\pi}{\omega}=\dfrac{2\pi~\text{rad}}{1.50~\text{rad/m}}=\boxed{4.18~\text{s}}$,
> - $f=\dfrac{1}{T}=\dfrac{1}{4.18~\text{s}}=\boxed{0.240~\text{Hz}}$.
4. What is the velocity of this wave?
> $v=\dfrac{\lambda}{T}=\dfrac{0.314~\text{m}}{4.18~\text{s}}=\boxed{0.0750~\text{m/s}}$

Now consider an element of string at position $x=12.0~\text{cm}$ and time $t=5.00~\text{s}$. Answer the questions 5–9.

5. What is of displacement of this string element?
>$\begin{align}y(0.12,5.00)&=0.00250\sin(20.0\times 0.12-1.50\times 5.00)\\&=0.00250\sin(-5.10)\\&=0.00250\times 0.926\\&=\boxed{0.00231~(\text{m})}\end{align}$
6. What is the **transverse velocity** $u\equiv\frac{\partial y}{\partial t}$ of this string element? (Do not confuse $u$ with $v$!)
>$u(x,t)\equiv\dfrac{\partial}{\partial t}y(x,t)=0.00250\times(-1.50)\times \cos(20.0x-1.50t)$
$\begin{align}u(0.12,5.00)&=0.00250\times(-1.50)\times \cos(20.0\times 0.12-1.50\times 5.00)\\&=\boxed{-0.00141~(\text{m/s})}\end{align}$
7. What is the transverse acceleration $a_y\equiv\frac{\partial^2 y}{\partial t^2}$ of this string element?
> $a_y(x,t)\equiv\dfrac{\partial}{\partial t}u(x,t)=-0.00250\times(-1.50)^2\times \sin(20.0x-1.50t)$
$\begin{align}a_y(0.12,5.00)&=-0.00250\times(-1.50)^2\times \sin(20.0\times 0.12-1.50\times 5.00)\\&=\boxed{-0.00520~\left(\text{m/s}^2\right)}\end{align}$
8. Using the results from 5–7, explain why this string element is moving (transversely) in simple harmonic motion.
> $\because -\omega^2y(x,t)=a_y(x,t)\quad\therefore$ it moves in simple harmonic motion.
9. Show that the wave function (1) satisfies the wave equation $$\dfrac{\partial^2 y}{\partial x^2}=\dfrac{1}{v^2}\dfrac{\partial^2 y}{\partial t^2}\tag{2}$$
> $\require{cancel}\left[\cancel{-0.00250}\times(20.0)^2\times \cancel{\sin(20.0x-1.50t)}\right]\\=\dfrac{1}{\left(0.0750\right)^2}\times\left[\cancel{-0.00250}\times(-1.50)^2\times \cancel{\sin(20.0x-1.50t)}\right],$ or $(20.0)^2=\dfrac{(-1.50)^2}{(0.0750)^2}$ for any $x$, $t$.

Now given that the string is under tension $\tau=45~\text{N}$, answer the questions 10–11.

10. What is the linear density of this string?
> $v=\sqrt{\dfrac{\tau}{\mu}}\implies \mu=\dfrac{\tau}{v^2}=\dfrac{45~\text{N}}{\left(0.0750~\text{m/s}\right)^2}=\boxed{1.5~\text{kg/m}}$
11. What is the average power of this string?
> $\begin{align}P_\text{avg}&=\dfrac{1}{2}\mu v\omega^2y_m^2=\dfrac{1}{2}\left(1.5~\text{kg/m}\right)\left(0.0750~\text{m/s}\right)\left(1.50~\text{rad/m}\right)^2\left( 0.00250~~\text{m}\right)^2\\&=\boxed{7.9\times 10^{-7}~\text{W}}\end{align}$

### Problem 2 (§16-9 – §16-11)
Two identical sinusoidal waves, moving in the same direction along a stretched string, interfere with each other. The amplitude $y_m$ of each wave is $8.0~\text{mm}$, and the phase difference $\phi$ between them is $75^\circ$.

1. What is the amplitude $y_m'$ of the resultant wave due to the interference?
> $y_m'=2y_m\cos\frac{\phi}{2}=2(8.0~\text{mm})\cos\left(\frac{75^\circ}{2}\right)=\boxed{13~\text{mm}}$.
2. What is the type of this interference? Fully constructive, fully destructive, or intermediate?
> intermediate.
3. What phase difference, in radians and wavelengths, will give the resultant wave an amplitude of $4.0~\text{mm}$?
> $y_m'=2y_m\cos\dfrac{\phi}{2}\implies\phi=2\cos^{-1}\dfrac{y_m'}{2y_m}=2\cos^{-1}\left(\dfrac{4.0~\text{mm}}{2\times 8.0~\text{mm}}\right)=\boxed{\pm 2.6~\text{rad}}$
> $\dfrac{\pm 2.6~\text{rad}}{2\pi~\text{rad}/\lambda}=\boxed{\pm 0.419\lambda}$

Now consider another two sinusoidal waves $y_1(x, t)$ and $y_2(x, t)$, with the same angular wave number $k$ and thus the same angular frequency $\omega$ as they travel together in the same direction along a string. Their amplitudes are $y_{m1}=10.0~\text{mm}$ and $y_{m2}=8.0~\text{mm}$, and their phase constants are $0$ and $\pi/3~\text{rad}$, respectively. For the resultant wave in the form of $$y'(x,t)=y'_m\sin(kx-\omega t+\beta),$$ then in questions 4–5, please use **phasors** to calculate:

4. the amplitude $y'_m$, and
5. the phase constant $\beta$.

### Problem 3 (§16-12 – §16-13)

Figure 16-22 shows a pattern of resonant oscillation of a string of mass $m=4.500~\text{g}$ and length $L=2.400~\text{m}$ and that is under tension $\tau=225.0~\text{N}$. Answer the following questions.
1. What is the wavelength $\lambda$ of the transverse waves producing the standing-wave pattern
2. what is the harmonic number $n$?
3. What is the frequency $f$ of the transverse waves and of the oscillations of the moving string elements?
4. What is the maximum magnitude of the transverse velocity $u_m$ of the element oscillating at coordinate $x=0.180~\text{m}$ (note the $x$ axis in the figure)?
5. At what point during the element’s oscillation is the transverse velocity maximum?