# 7.6: Other Angular Counterparts

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# Work

We are familiar with the concept of work as a way that the energy of a system is changed. In terms of force and distance, work is:

\[ W = \int F_{\parallel}dx \]

where the parallel symbol reminds us that it is only the components of force and displacement in the same direction that contribute to the integral.

A similar expression holds for the work done by a torque which acts through an angle:

\[ W = \int \tau_{\parallel}d\theta \]

The energy of a particular system can be changed by the process of a force exerted by an outside object doing work on an object in the system and/or by a torque exerted by an outside object doing work on an object within the system. In either case, the work can be positive (increases the energy of the system) or negative (decreases the energy of the system). If the force or torque is constant (or we assume an average force or torque), the integral is immediately performed and we have

\[ W = F_{\parallel}\Delta x \]

and

\[ W = \tau_{\parallel}\Delta\theta \]

# Energy Systems

The total energy of a system is the sum of all of the various energy systems, which can include both translational and rotational energy systems. During collisions among parts of a physical system, energy can be transferred among these separate systems. We have previously mentioned rotational kinetic energy. Another energy system with a rotational counterpart is elastic or spring potential energy. The elastic potential energy of a system described by a spring constant k is:

\[ PE_{elastic}=\frac{1}{2}kx^2 \]

Similarly, the elastic potential energy of a rotating system which has a linear restoring force is given by the expression:

\[ PE_{elastic} = \frac{1}{2}k\theta^2 \]

# The rate of Energy Transfer: Power

We previously discussed power as the time derivative of energy transfer, or the rate at which the energy of a system changes. This applies, of course, to any type of energy system. Recall that the SI unit of power is the watt (W) which is equal to a joule per second.

In mechanical systems, in which energy is transferred as work, it is often useful to consider the rate of energy transfer, power, associated with a particular force. Since the energy transferred is the work done by the force, the power associated with that force is the time derivative of the work. If the force is constant in time, then:

\[ P = \dfrac{dW}{dt} = \dfrac{d(F_{\parallel avg}x)}{dt}=F_{\parallel avg} \dfrac{dx}{dt}=F_{\parallel avg} v \]

Thus, the power is simply the applied force times the velocity of the object the force is acting on. The rotational counterpart is:

\[P = \tau_{\parallel}\omega\]

# Putting it all together

The chart on below shows all of the linear motion and dynamic variables along with their rotational counterparts. Keep this chart out and handy for ready reference to help you from getting “lost” in all the symbols. You should make sure that you recognize the meaning behind the symbols when you see on of these relationships. (Note: acceleration, angular acceleration, and Newton’s second law are treated in detail in the next chapter, but are shown here for completeness and convenience)

Category | Concept | Translation | Rotation | Relation |
---|---|---|---|---|

Kinematic Variables | Position, Velocity, Acceleration | \(x\), \(v=dx/dt\), \(a=dv/dt\) | \(\theta\), \(\omega=d\theta /dt\), \(\alpha =d\omega /dt\) | \(\theta=arclength/r\) \(\omega=v_t/r\) \(\alpha=a_t/r\) |

Fundamental Dynamic Variables | Force/Torque, Inertia, Momentum | \(F\) \(m\) \(p=mv\) | \(\tau\) \(I\) \(L=I\omega\) | \(\tau=r_{\perp}F=rF_{tang}\) \(I=\sum mr^2\) \(L=r_{perp}p=rp_{tang}\) |

Energy | Elastic Energy Kinetic Energy Work Energy Conversation Power System Energy | \(1/2~ kx^2\) \(1/2 ~mv^2\) \(W=\int F_{\parallel}dx\) \(\Delta E_{system}=W_{all}=Q\) \(P=dE/dt=F _{\parallel}v\) All Kinetic and Potential Energies plus Thermal | \(1/2~k\theta^2\) \(1/2 ~I\omega^2\) \(W=\int \tau_{\parallel}d\theta\) \(\Delta E_{system}=W_{all}=Q\) \(P=dE/dt=\tau _{\parallel}\omega\) All Kinetic and Potential Energies plus Thermal | |

Momentum | Momentum Impulse Momentum Conservation | \(p=mv\) \(J=\int Fdt\) \(\Delta p=J_{ext}\) | \(L=I\omega\) \(AngJ=\int \tau dt\) \(\Delta L=angJ_{ext}\) | \(L=r_{perp}p=rp_{tang}\) |

Newton's Laws | Newton's 1^{st} Law Newton's 2^{nd} Law Newton's 3^{rd} Law | \(if~ \sum F=0, then~\Delta v=0, \Delta p=0 \) \(\sum F=ma ~or,\sum F=dp/dt\) \(F_{1~on~2}=-F_{2~on~1}\) \(J_{1~on~2}=-J_{2~on~1}\) | \(if~ \sum \tau=0, then~\Delta L=0\) \(\sum \tau=I\alpha ~or,\sum \tau=dL/dt\) |