Circular Motion and Angular Dynamics

Flashcards covering definitions, general concepts, assumptions, equations, and applications of circular and angular motion from Chapter 13.

What defines circular motion?

The motion of an object in a circular path.

Why is circular motion also referred to as angular motion?

Because the angle subtended by an object about a certain point continuously changes during circular motion.

Besides circular motion, what are two other types of angular motion?

Rotatory motion and motion in an elliptical path.

What key assumption must be met by a body in angular motion so that all of its particles move together?

The body should be rigid.

In a rigid body undergoing angular motion, what three angular quantities are identical for all particles regardless of their distance from the axis?

All particles have the same angular displacement ($\theta$), the same angular velocity ($\omega$), and the same angular acceleration ($\alpha$).

Why is the axis of rotation assumed to be fixed in angular motion calculations?

A fixed axis restricts angular quantities to only two directions (up and down), making calculations simpler.

What are angular analogues?

Corresponding angular variables that replace linear variables in angular motion (also called moments), functioning in the same way as linear variables.

What is the angular analogue of linear force, and what are their respective SI units?

The angular analogue of force is torque. The unit of force is newton ($\text{N}$) and the unit of torque is newton meter ($\text{N\,m}$).

What physical quantity acts as the rotational analogue of linear inertia?

Moment of inertia ($I$).

How is the direction of axial angular quantities determined using the Right Hand Rule?

Curl the fingers of your right hand in the direction of rotation; the erected thumb points in the direction of the axis.

What rule governs the direction of angular acceleration relative to the axis of rotation?

Angular acceleration is directed along the axis if the object is speeding up, and opposite to the axis if the object is slowing down.

What are the three equations of motion for angular motion with constant angular acceleration?

1. $\omega_f = \omega_i + \alpha t$
2. $\theta = \omega_i t + \frac{1}{2} \alpha t^2$
3. $2\alpha\theta = \omega_f^2 - \omega_i^2$

How are tangential, axial, and radial quantities defined in circular motion?

Tangential quantities are directed along the tangent; axial quantities are directed along the axis of rotation; radial quantities are directed parallel to the radius from the circumference towards the center.

According to Advanced Concept 13.1, which vector quantities are never constant in circular motion?

Tangential vectors and radial vectors are never constant in circular motion. Axial vectors may or may not be constant.

What is angular displacement, and when is it treated as a vector quantity?

Angular displacement ($\theta$) is the angle swept by a particle moving in angular motion in a time interval. It is a scalar for large angles, but considered a vector quantity for small angles.

What is the SI unit and dimension of angular displacement?

The SI unit is radian ($\text{rad}$), and it is a dimensionless quantity ($[\text{L}^0 \text{L}^0]$).

What are the primary conversion values between degrees, radians, and revolutions?

$360^\circ = 2\pi\,\text{rad}$, $1^\circ = \frac{\pi}{180}\,\text{rad} \approx 0.0174\,\text{rad}$, $1\,\text{rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ$, and $1\,\text{rev} = 2\pi\,\text{rad} = 360^\circ$.

According to the Screw Rule, what directions correspond to positive and negative angular displacement?

Clockwise rotation corresponds to negative angular displacement directed into the plane of paper (represented by a cross $\times$); anticlockwise rotation corresponds to positive angular displacement directed out of the plane of paper (represented by a dot $\cdot$).

What is angular velocity, and what are its SI unit and dimensions?

Angular velocity is the rate of change of angular displacement ($\omega = \frac{\Delta \theta}{\Delta t}$). Its SI unit is $\text{rad\,s}^{-1}$ and its dimension is $[\text{T}^{-1}]$.

How is $1\,\text{rpm}$ (revolution per minute) converted into $\text{rad\,s}^{-1}$?

$1\,\text{rpm} = \frac{2\pi\,\text{rad}}{60\,\text{s}} = \frac{\pi}{30}\,\text{rad\,s}^{-1}$.

What are the angular velocities of Earth around the Sun and around its own axis?

Around the Sun: $2\pi\,\text{rad/year}$; around its own axis: 2\n\pi\,\text{rad/day}.

What are the speeds of the Second Hand, Minute Hand, and Hour Hand of a clock in $\text{rad\,s}^{-1}$?

Second Hand: $\frac{\pi}{30}\,\text{rad\,s}^{-1}$; Minute Hand: $\frac{\pi}{1800}\,\text{rad\,s}^{-1}$; Hour Hand: $\frac{\pi}{21600}\,\text{rad\,s}^{-1}$.

What is the ratio of angular velocities of the clock hands (Second Hand : Minute Hand : Hour Hand)?

$1 : \frac{1}{60} : \frac{1}{720}$.

When an object moves with uniform angular velocity, how does its average angular velocity compare to its instantaneous angular velocity?

Its average angular velocity is equal to its instantaneous angular velocity.

What is angular acceleration, and what causes it?

Angular acceleration ($\alpha = \frac{\Delta \omega}{\Delta t}$) is the rate of change of angular velocity. It is caused by torque ($\tau = I \alpha$).

What is the SI unit and dimension of angular acceleration?

The SI unit is $\text{rad\,s}^{-2}$ and its dimension is $[\text{T}^{-2}]$.

What are the vector equations relating linear displacement, linear velocity, and linear acceleration to their angular analogues?

Linear displacement: $\mathbf{s} = \boldsymbol{\theta} \times \mathbf{r}$; Linear velocity: $\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}$; Linear acceleration: $\mathbf{a} = \boldsymbol{\alpha} \times \mathbf{r}$.

What is centripetal force and in what direction does it point?

Centripetal force is the force that causes an object to move in a circular path instead of a straight path; it is always directed toward the center of the circle.

What are the formula expressions for centripetal acceleration ($a_c$)?

$a_c = \frac{v^2}{r} = \omega^2 r = 4\pi^2 f^2 r = \frac{4\pi^2 r}{T^2}$.

What are the formulas for centripetal force ($F_c$) in terms of kinetic energy ($\text{KE}$) and linear momentum ($p$)?

In terms of kinetic energy: $F_c = \frac{2\text{KE}}{r}$; in terms of linear momentum: $F_c = \frac{p^2}{m r}$.

What is the vector expression for centripetal force ($\mathbf{F}_c$) using the radial vector $\mathbf{r}$?

$\mathbf{F}_c = -m \omega^2 \mathbf{r} = -m \omega^2 r \hat{\mathbf{r}}$, where the minus sign indicates the direction is toward the center, opposite to the radial vector $\mathbf{r}$.

What provides the centripetal force for a satellite orbiting Earth, and what is its orbital velocity formula?

The force of gravity ($F_g$) provides the centripetal force, giving an orbital velocity of $v = \sqrt{\frac{GM}{r}}$.

What provides the centripetal force for a car making a turn on an unbanked circular road?

The frictional force between the tires and the road ($F_c = F_f = \mu N = \mu m g$).

What formula gives the maximum safe speed $v$ achievable on a banked road without skidding?

$v = \sqrt{r g \tan(\theta)}$.

Why do centripetal and centrifugal forces not constitute an action-reaction pair under Newton's third law?

Because action-reaction pairs must act on two different bodies, whereas both centripetal and centrifugal forces appear to act on the same object performing circular motion.

How does gravitational acceleration $g$ vary across Earth's surface from the poles to the equator?

The value of $g$ is maximum at the poles (shorter distance to Earth's center) and minimum at the equator (longer distance and Earth's rotation). Moving from poles to equator decreases $g$.

What are the formulas for gravitational acceleration $g'$ at height $h$ above Earth's surface and depth $d$ inside Earth?

At height $h$: $g' = \frac{GM}{(R+h)^2}$; at depth $d$: $g' = \frac{GM(R-d)}{R^3}$.

In which four scenarios is the gravitational acceleration $g$ equal to zero?

1. At the center of any planet.
2. At an infinite distance from the planet.
3. Inside satellites revolving around the earth.
4. Inside a freely falling system.