PHY 151 Quiz 4: Oscillations, Thermal Expansion, Heat Transfer, and Calorimetry – Step-by-Step Study Guidance

Q1. A mass on a spring oscillates horizontally with position given by $x(t) = (6\ \text{cm}) \cos[(12.57\ \text{rad/s})\ t]$.

Background

Topic: Simple Harmonic Motion (SHM)

This question tests your understanding of the mathematical description of SHM, including amplitude, period, and frequency, as well as how to use the position function to find the mass's location at a given time.

Key Terms and Formulas:

• Amplitude ($A$): The maximum displacement from equilibrium.

• Angular frequency ($\omega$): The coefficient of $t$ inside the cosine, in rad/s.

• Period ($T$): $T = \frac{2\pi}{\omega}$

• Frequency ($f$): $f = \frac{1}{T}$

• Position at time $t$: $x(t) = A \cos(\omega t)$

Step-by-Step Guidance

1. Amplitude: Identify the amplitude $A$ from the equation $x(t) = (6\ \text{cm}) \cos[(12.57\ \text{rad/s})\ t]$. The amplitude is the coefficient in front of the cosine.

2. Period and Frequency: The angular frequency $\omega$ is $12.57\ \text{rad/s}$. Use the formulas:

   • $T = \frac{2\pi}{\omega}$

   • $f = \frac{1}{T}$

   Plug in the value of $\omega$ to find $T$ and then $f$.

3. Position at $t = 0.4$ s: Substitute $t = 0.4$ s into the position equation:

   • $x(0.4) = (6\ \text{cm}) \cos[(12.57\ \text{rad/s}) \times 0.4\ \text{s}]$

   Calculate the argument of the cosine, then evaluate the cosine, and multiply by 6 cm.

4. For the graph labeling parts, remember that amplitude is the maximum value on the $x$-axis, and one period is the time it takes for the motion to repeat.

Try solving on your own before revealing the answer!

Final Answers:

• Amplitude: 6 cm

• Period: 0.5 s

• Frequency: 2.0 Hz

• Position at $t = 0.4$ s: 1.86 cm

These values are found by applying the SHM formulas and substituting the given numbers. The amplitude is the coefficient, the period and frequency come from the angular frequency, and the position is calculated by evaluating the cosine at the specified time.

Q2. A concrete slab (10 cm × 20 cm × 2 cm) hardens at 22°C. Calculate the change in width and the rate of heat transfer on a hot day (49°C).

Background

Topic: Thermal Expansion and Heat Transfer

This question tests your ability to apply the concepts of linear thermal expansion and heat conduction through a solid.

Key Terms and Formulas:

• Linear Expansion: $\Delta L = \alpha L_i \Delta T$

• Thermal Conductivity (Heat Transfer Rate): $\frac{Q}{\Delta t} = \frac{kA}{L} \Delta T$

• $\alpha$ = coefficient of linear expansion, $k$ = thermal conductivity, $A$ = area, $L$ = thickness, $\Delta T$ = temperature difference

Step-by-Step Guidance

1. Change in Width:

   • Identify $L_i$ (initial width), $\alpha$ (given), and $\Delta T$ (temperature change).

   • Calculate $\Delta T = 49^\circ\text{C} - 22^\circ\text{C}$.

   • Plug values into $\Delta L = \alpha L_i \Delta T$ to find the change in width.

2. Direction of Change: Since temperature increases, the width will increase (thermal expansion).

3. Heat Transfer Rate:

   • Identify $k$ (thermal conductivity), $A$ (area = length × width), $L$ (thickness), and $\Delta T$ (temperature difference).

   • Plug values into $\frac{Q}{\Delta t} = \frac{kA}{L} \Delta T$ to set up the calculation.

Try solving on your own before revealing the answer!

Final Answers:

• Change in width: $7.0 \times 10^{-5}$ m (width increases)

• Heat transfer rate: 68 W

Both answers are found by substituting the given values into the respective formulas for linear expansion and heat conduction.

Q3. Galileo thermometer: As the liquid cools, what happens to its volume and density?

Background

Topic: Thermal Expansion and Density

This question tests your understanding of how temperature changes affect the volume and density of liquids.

Key Terms and Concepts:

• Thermal Expansion: As temperature decreases, most liquids contract (volume decreases).

• Density ($\rho$): $\rho = \frac{m}{V}$; as volume decreases (with constant mass), density increases.

Step-by-Step Guidance

1. As the liquid cools, consider what happens to the volume due to thermal contraction.

2. Recall that density is inversely related to volume for a fixed mass.

3. Choose the correct options: volume (increase, decrease, stay the same); density (increase, decrease, stay the same).

Try solving on your own before revealing the answer!

Final Answer:

As the liquid cools, its volume decreases and its density increases.

This is because cooling causes contraction (smaller volume), and with the same mass, density must go up.

Q4. A hot chunk of silver is placed in water. The system reaches thermal equilibrium. (A) Draw energy flow diagrams. (B) Calculate the mass of water.

Background

Topic: Calorimetry (Heat Transfer and Thermal Equilibrium)

This question tests your ability to analyze energy flow in a calorimetry experiment and to use the principle of conservation of energy to solve for unknown quantities.

Key Terms and Formulas:

• Heat transfer: $Q = mc\Delta T$

• Thermal equilibrium: $|Q_{\text{water}}| = |Q_{\text{silver}}|$

• Specific heat ($c$): Amount of heat required to change temperature of 1 kg by 1 K.

Step-by-Step Guidance

1. Energy Flow Diagrams: Draw bar graphs for the thermal energy ($E_{th}$) and phase energy ($E_{ph}$) for both water and silver, before and after mixing. Show that $E_{th}$ increases for water and decreases for silver, and both reach the same final temperature.

2. Set up the energy balance equation:

   • $|Q_{\text{water}}| = |Q_{\text{silver}}|$

   • Write $Q_{\text{water}} = m_w c_w (T_f - T_{w,i})$

   • Write $Q_{\text{silver}} = m_{Ag} c_{Ag} (T_f - T_{Ag,i})$

3. Solve for the unknown mass of water ($m_w$): Rearrange the equation to solve for $m_w$ in terms of the other variables.

4. Plug in the given values: Substitute the values for $m_{Ag}$, $c_{Ag}$, $T_f$, $T_{Ag,i}$, $c_w$, and $T_{w,i}$ into your equation for $m_w$.

Try solving on your own before revealing the answer!

Final Answer:

The mass of water is 0.237 kg.

This is found by equating the heat lost by silver to the heat gained by water and solving for $m_w$ using the specific heats and temperature changes provided.