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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 14.7 Practice - Rotational Energy and Angular Momentum

Subsection A*R*C*S Practice

A*R*C*S 14.7.1. The Merry-Go-Round I.

The Sun has mass \(>2 \times 10^{30} \mathrm{~kg}\text{,}\) radius \(7\times 10^{5} \mathrm{~km}\text{,}\) and rotational period of approximately 28 days. If the Sun should collapse into a white dwarf of radius \(3.5 \times 10^3 \mathrm{~km}\text{,}\) what would its period be if no mass were ejected and a sphere of uniform density can model the Sun both before and after?
Solution.
For this system, angular momentum is conserved since there will be no torque that acts on the system. This means the initial angular momentum equals the final angular momentum: \(L_i = L_f\text{.}\) You can use the moment of inertia of a solid uniform sphere, and relate the angular speed to the period of the rotation:
\begin{equation*} \frac{2}{5} M_s R_s ^2 (\frac{2\pi}{T_s}) = \frac{2}{5} M_s R_n ^2 (\frac{2\pi}{T_n}) \end{equation*}
Solving for the new time:
\begin{equation*} T_n = T_s \frac{R_n^2}{R_s^2} = (28 \ \mathrm{~days})\frac{(3.5\times10^3)^2}{(7\times10^5)^2} = 0.0007 \ \mathrm{~days} = 60.5 \mathrm{~s} \end{equation*}

A*R*C*S 14.7.2. The Merry-Go-Round I.

You are standing at the edge of a merry-go-round at the playground. You and the merry-go-round have about the same mass, and you can treat the merry-go-round as a solid disc with moment of inertia \(I = \frac{1}{2}mR^2\text{,}\) where \(R\) is the radius. The merry-go-round is initially spinning with constant angular speed \(\omega_i\text{.}\) You decide to jump off the merry-go-round; you jump in a horizontal plane tangent to its edge.
The merry-go-round comes to a complete stop. What is your speed immediately after leaving the merry-go-round?

Subsection Explanation Practice

Explanation 14.7.3. The Merry-Go-Round II.

Consider the system of both you and the merry-go-round from the previous activity. Did the kinetic energy of this system increase, decrease, or stay the same as a result of you jumping off?

Subsection Numerical Practice

Calculation 14.7.4. Olympic High Diver.

An Olympic high diver in midair pulls her legs inward toward her chest. Doing so changes which of these quantities?
  1. Angular momentum
  2. Rotational inertia about her center of mass
  3. Angular velocity
  4. Translational (linear) momentum
  5. Translational (linear) kinetic energy
  6. Rotational kinetic energy
Answer.
B., C., E.

Calculation 14.7.5. Helicopter I.

A typical small rescue helicopter has four blades. Each blade is \(4.0 \mathrm{~m}\) long and has a mass of \(50.0 \mathrm{~kg}\text{.}\) The blades can be approximated as thin rods that rotate about one end of an axis perpendicular to their length. The helicopter has a total loaded mass of \(1000 \mathrm{~kg}\text{.}\) Calculate the rotational kinetic energy in the blades when they rotate at \(300 \mathrm{~rpm}\text{.}\)
Answer.
\(5.26 \times 10^5 \mathrm{~J}\)

Calculation 14.7.6. Helicopter II.

A typical small rescue helicopter has four blades. Each blade is \(4.00 \mathrm{~m}\) long and has a mass of \(50.0 \mathrm{~kg}\text{.}\) The blades can be approximated as thin rods that rotate about one end of an axis perpendicular to their length. The helicopter has a total loaded mass of \(1000 \mathrm{~kg}\text{.}\) What is the ratio of translational kinetic energy of the helicopter over the rotational kinetic energy of its blades when it flies at \(20.0 \mathrm{~m/s}\text{?}\)
Answer.
0.38

Calculation 14.7.7. Helicopter III.

A typical small rescue helicopter has four blades. Each blade is \(4.00 \mathrm{~m}\) long and has a mass of \(50.0 \mathrm{~kg}\text{.}\) The blades can be approximated as thin rods that rotate about one end of an axis perpendicular to their length. The helicopter has a total loaded mass of \(1000 \mathrm{~kg}\text{.}\) To what height could the helicopter be raised if all of the rotational kinetic energy could be used to lift it?
Answer.
\(53.7 \mathrm{~m}\)

References References

[1]
Numerical practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.
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