| Step | Reasoning |
|---|---|
| Identify that the internal work done by the student is equal to the change in the system’s rotational kinetic energy. \[ W = \Delta K = K_f – K_0 \] |
According to the work-energy theorem, the total work done on a system is equal to its change in kinetic energy. In this case, the student performs internal work to move the weights. |
| Determine the final angular velocity \(\omega_f\) using the conservation of angular momentum. \[ L_0 = L_f \implies I_0 \omega_0 = I_f \omega_f \implies \omega_f = \dfrac{I_0 \omega_0}{I_f} \] |
Since the platform rotates with negligible friction and the forces used to pull the weights are internal to the system, there is no net external torque. Therefore, angular momentum \(L\) is conserved. |
| Substitute \(\omega_f\) into the expression for final rotational kinetic energy \(K_f\). \[ K_f = \dfrac{1}{2} I_f \omega_f^2 = \dfrac{1}{2} I_f \left( \dfrac{I_0 \omega_0}{I_f} \right)^2 = \dfrac{I_0^2 \omega_0^2}{2 I_f} \] |
To calculate the change in energy, we need the final kinetic energy in terms of the given initial variables. |
| Calculate the work done by finding the difference between the final and initial kinetic energies and simplifying. \[ W = \dfrac{I_0^2 \omega_0^2}{2 I_f} – \dfrac{1}{2} I_0 \omega_0^2 = \dfrac{I_0 \omega_0^2}{2} \left( \dfrac{I_0}{I_f} – 1 \right) \] |
By subtracting the initial kinetic energy from the derived final energy, we find the work done by the student. |
Why each choice is correct or incorrect:
(A) This expression assumes that angular velocity is constant, which is incorrect because the reduction in rotational inertia must lead to an increase in angular velocity to conserve angular momentum.
(B) This is the correct answer derived from the combination of the work-energy theorem and angular momentum conservation.
(C) This expression correctly identifies how angular velocity changes but incorrectly uses the initial rotational inertia \(I_0\) in the kinetic energy formula for the final state.
(D) This expression assumes kinetic energy is directly proportional to rotational inertia (treating \(\omega\) as constant), which implies the energy decreases, even though the student must do work against centripetal forces to pull the weights inward.
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A car travels along a straight road between two cities separated by a total distance of \(2D\). The car travels the first distance \(D\) at a constant speed \(v_0\) and the remaining distance \(D\) at a constant speed \(3v_0\). Which of the following correctly identifies the average speed \(v_{avg}\) of the car for the entire trip and provides a valid justification?
A student conducts a laboratory experiment where a cart is moved \(2.0 \text{ m}\) to the right and then \(1.0 \text{ m}\) to the left along a straight, horizontal track. The student calculates the total distance traveled and the final displacement of the cart. Which of the following correctly classifies these quantities and provides a valid justification?
An experimental automated cart is tested on a linear track. A computer-controlled sensor measures the cart’s velocity \(v\) as a function of time \(t\), as shown in the graph. What is the displacement of the cart during the time interval from \(t = 0 \text{ s}\) to \(t = 10 \text{ s}\)?
A hiker starts at a trailhead and walks \(3.0 \text{ km}\) due North. The hiker then turns and walks \(5.0 \text{ km}\) in a direction \(37^\circ\) South of East to reach a campsite. (Note: \(\sin 37^\circ \approx 0.60\); \(\cos 37^\circ \approx 0.80\)). What is the magnitude of the hiker’s total displacement from the trailhead to the campsite?
A laboratory cart is restricted to motion along a horizontal track. A motion sensor records the direction of the cart’s velocity and the direction of its acceleration at three different times, as shown in the table below.
| Time | Direction of Velocity | Direction of Acceleration |
| :— | :— | :— |
| \(t_1\) | Right | Left |
| \(t_2\) | Left | Left |
| \(t_3\) | Left | Right |
Which of the following correctly describes the motion of the cart at each time?
A test rocket moves along a straight, horizontal track. A sensor records the rocket’s acceleration as a function of time, as shown in the graph below.
What is the average acceleration of the rocket during the time interval from \(t = 0 \text{ s}\) to \(t = 5 \text{ s}\)?
A sprinter crosses the finish line of a race moving with a velocity of \(12 \text{ m/s}\). The sprinter continues to run at this constant velocity for a reaction time of \(1.0 \text{ s}\) before beginning to slow down with a constant acceleration. If the sprinter comes to a complete stop exactly \(4.0 \text{ s}\) after crossing the finish line, what is the magnitude of the sprinter’s acceleration during the braking phase?
A surveyor starts at point \(P\) and walks a distance \(d\) due north. The surveyor then turns and walks an equal distance \(d\) in a direction \(60^{\circ}\) west of north to reach point \(Q\). What is the magnitude of the surveyor’s total displacement from point \(P\) to point \(Q\)?
A train is traveling at a constant speed \( v_0 \) when the engineer applies the brakes, resulting in a constant deceleration of magnitude \( a_0 \) that brings the train to a stop in a distance \( d \). In a second trial, the train is traveling at a speed \( 2v_0 \) when the brakes are applied and is brought to a stop with a constant deceleration of magnitude \( 2a_0 \). Which of the following is the stopping distance for the train in the second trial?
A small experimental vehicle starts from rest and accelerates with a constant acceleration \(a\) over a horizontal distance \(d\). After this distance, the vehicle’s engine is adjusted such that it continues to accelerate at a constant rate of \(\dfrac{a}{2}\) for an additional horizontal distance \(d\). Which of the following expressions represents the speed of the vehicle after it has traveled the total distance \(2d\)?
B
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| Kinematics | Forces |
|---|---|
| \(\Delta x = v_i t + \frac{1}{2} at^2\) | \(F = ma\) |
| \(v = v_i + at\) | \(F_g = \frac{G m_1 m_2}{r^2}\) |
| \(v^2 = v_i^2 + 2a \Delta x\) | \(f = \mu N\) |
| \(\Delta x = \frac{v_i + v}{2} t\) | \(F_s =-kx\) |
| \(v^2 = v_f^2 \,-\, 2a \Delta x\) |
| Circular Motion | Energy |
|---|---|
| \(F_c = \frac{mv^2}{r}\) | \(KE = \frac{1}{2} mv^2\) |
| \(a_c = \frac{v^2}{r}\) | \(PE = mgh\) |
| \(T = 2\pi \sqrt{\frac{r}{g}}\) | \(KE_i + PE_i = KE_f + PE_f\) |
| \(W = Fd \cos\theta\) |
| Momentum | Torque and Rotations |
|---|---|
| \(p = mv\) | \(\tau = r \cdot F \cdot \sin(\theta)\) |
| \(J = \Delta p\) | \(I = \sum mr^2\) |
| \(p_i = p_f\) | \(L = I \cdot \omega\) |
| Simple Harmonic Motion | Fluids |
|---|---|
| \(F = -kx\) | \(P = \frac{F}{A}\) |
| \(T = 2\pi \sqrt{\frac{l}{g}}\) | \(P_{\text{total}} = P_{\text{atm}} + \rho gh\) |
| \(T = 2\pi \sqrt{\frac{m}{k}}\) | \(Q = Av\) |
| \(x(t) = A \cos(\omega t + \phi)\) | \(F_b = \rho V g\) |
| \(a = -\omega^2 x\) | \(A_1v_1 = A_2v_2\) |
| Constant | Description |
|---|---|
| [katex]g[/katex] | Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface |
| [katex]G[/katex] | Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex] |
| [katex]\mu_k[/katex] and [katex]\mu_s[/katex] | Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion. |
| [katex]k[/katex] | Spring constant, in [katex]\text{N/m}[/katex] |
| [katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex] | Mass of the Earth |
| [katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex] | Mass of the Moon |
| [katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex] | Mass of the Sun |
| Variable | SI Unit |
|---|---|
| [katex]s[/katex] (Displacement) | [katex]\text{meters (m)}[/katex] |
| [katex]v[/katex] (Velocity) | [katex]\text{meters per second (m/s)}[/katex] |
| [katex]a[/katex] (Acceleration) | [katex]\text{meters per second squared (m/s}^2\text{)}[/katex] |
| [katex]t[/katex] (Time) | [katex]\text{seconds (s)}[/katex] |
| [katex]m[/katex] (Mass) | [katex]\text{kilograms (kg)}[/katex] |
| Variable | Derived SI Unit |
|---|---|
| [katex]F[/katex] (Force) | [katex]\text{newtons (N)}[/katex] |
| [katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy) | [katex]\text{joules (J)}[/katex] |
| [katex]P[/katex] (Power) | [katex]\text{watts (W)}[/katex] |
| [katex]p[/katex] (Momentum) | [katex]\text{kilogram meters per second (kgm/s)}[/katex] |
| [katex]\omega[/katex] (Angular Velocity) | [katex]\text{radians per second (rad/s)}[/katex] |
| [katex]\tau[/katex] (Torque) | [katex]\text{newton meters (Nm)}[/katex] |
| [katex]I[/katex] (Moment of Inertia) | [katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex] |
| [katex]f[/katex] (Frequency) | [katex]\text{hertz (Hz)}[/katex] |
Metric Prefixes
Example of using unit analysis: Convert 5 kilometers to millimeters.
Start with the given measurement: [katex]\text{5 km}[/katex]
Use the conversion factors for kilometers to meters and meters to millimeters: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}[/katex]
Perform the multiplication: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}[/katex]
Simplify to get the final answer: [katex]\boxed{5 \times 10^6 \, \text{mm}}[/katex]
Prefix | Symbol | Power of Ten | Equivalent |
|---|---|---|---|
Pico- | p | [katex]10^{-12}[/katex] | |
Nano- | n | [katex]10^{-9}[/katex] | |
Micro- | µ | [katex]10^{-6}[/katex] | |
Milli- | m | [katex]10^{-3}[/katex] | |
Centi- | c | [katex]10^{-2}[/katex] | |
Deci- | d | [katex]10^{-1}[/katex] | |
(Base unit) | – | [katex]10^{0}[/katex] | |
Deca- or Deka- | da | [katex]10^{1}[/katex] | |
Hecto- | h | [katex]10^{2}[/katex] | |
Kilo- | k | [katex]10^{3}[/katex] | |
Mega- | M | [katex]10^{6}[/katex] | |
Giga- | G | [katex]10^{9}[/katex] | |
Tera- | T | [katex]10^{12}[/katex] |
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