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HomeStudy GuidesPhysics

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The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

Introduction to Work, Energy, and Energy ResourcesWork: The Scientific DefinitionKinetic Energy and the Work-Energy TheoremGravitational Potential EnergyVideo: Potential and Kinetic EnergyConservative Forces and Potential EnergyNonconservative ForcesConservation of EnergyPowerWork, Energy, and Power in HumansWorld Energy Use

Linear Momentum and Collisions

Rotational Motion and Angular Momentum

Introduction to Rotational Motion and Angular MomentumAngular AccelerationKinematics of Rotational MotionVideo: Rotational MotionDynamics of Rotational Motion: Rotational InertiaRotational Kinetic Energy: Work and Energy RevisitedAngular Momentum and Its ConservationVideo: Angular MomentumCollisions of Extended Bodies in Two DimensionsGyroscopic Effects: Vector Aspects of Angular Momentum

Statics and Torque

Fluid Statics

Introduction to Fluid StaticsWhat Is a Fluid?DensityPressureVariation of Pressure with Depth in a FluidPascal's PrincipleGauge Pressure, Absolute Pressure, and Pressure MeasurementArchimedes' PrincipleVideo: BuoyancyCohesion and Adhesion in Liquids: Surface Tension and Capillary ActionPressures in the Body

Fluid Dynamics and Its Biological and Medical Applications

Introduction to Fluid Dynamics and Biological and Medical ApplicationsFlow Rate and Its Relation to VelocityBernoulli's EquationVideo: Fluid FlowThe Most General Applications of Bernoulli's EquationViscosity and Laminar Flow; Poiseuille's LawThe Onset of TurbulenceMotion of an Object in a Viscous FluidMolecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

Temperature, Kinetic Theory, and the Gas Laws

Heat and Heat Transfer Methods

Thermodynamics

Introduction to ThermodynamicsThe First Law of ThermodynamicsThe First Law of Thermodynamics and Some Simple ProcessesIntroduction to the Second Law of Thermodynamics: Heat Engines and Their EfficiencyCarnot's Perfect Heat Engine: The Second Law of Thermodynamics RestatedApplications of Thermodynamics: Heat Pumps and RefrigeratorsEntropy and the Second Law of Thermodynamics: Disorder and the Unavailability of EnergyStatistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

Oscillatory Motion and Waves

Introduction to Oscillatory Motion and WavesHooke's Law: Stress and Strain RevisitedPeriod and Frequency in OscillationsSimple Harmonic Motion: A Special Periodic MotionVideo: Harmonic MotionThe Simple PendulumEnergy and the Simple Harmonic OscillatorUniform Circular Motion and Simple Harmonic MotionDamped Harmonic MotionForced Oscillations and ResonanceWavesSuperposition and InterferenceEnergy in Waves: Intensity

Physics of Hearing

- Describe uniform circular motion.
- Explain non-uniform circular motion.
- Calculate angular acceleration of an object.
- Observe the link between linear and angular acceleration.

Uniform Circular Motion and Gravitation discussed only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity *ω* was defined as the time rate of change of angle *θ*:

$\omega =\frac{\Delta \theta}{\Delta t}\\$

$\omega =\frac{v}{r}\\$

where $\alpha =\frac{\Delta \omega }{\Delta t}\\$

,
Suppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the angular acceleration in rad/s^{2}. (b) If she now slams on the brakes, causing an angular acceleration of -87.3 rad/s^{2}, how long does it take the wheel to stop?

**Strategy for (a)**

The angular acceleration can be found directly from its definition in *ω* is 250 rpm and Δ*t* is 5.00 s.

**Solution for (a)**

Entering known information into the definition of angular acceleration, we get

*ω* is in revolutions per minute (rpm) and we want the standard units of rad/s^{2} for angular acceleration, we need to convert Δ*ω* from rpm to rad/s:

*α*, we get

**Strategy for (b)**

In this part, we know the angular acceleration and the initial angular velocity. We can find the stoppage time by using the definition of angular acceleration and solving for Δ*t*, yielding

**Solution for (b)**

Here the angular velocity decreases from 26.2 rad/s (250 rpm) to zero, so that Δ*ω* is –26.2 rad/s, and *α* is given to be -87.3 rad/s^{2}. Thus,

**Discussion**

Note that the angular acceleration as the girl spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is large and negative. The angular velocity quickly goes to zero. In both cases, the relationships are analogous to what happens with linear motion. For example, there is a large deceleration when you crash into a brick wall—the velocity change is large in a short time interval.

$\alpha =\frac{\Delta \omega }{\Delta t}\\$

because the final angular velocity and time are given. We see that Δ$\begin{array}{lll}\alpha & =& \frac{\Delta \omega }{\Delta t}\\ & =& \frac{\text{250 rpm}}{\text{5.00 s}}\text{.}\end{array}\\$

Because Δ$\begin{array}{c}\Delta{\omega} &=& 250 \frac{\text{rev}}{\text{min}} \cdot \frac{2\pi\text{ rad}}{\text{rev}} \cdot \frac{1\text{ min}}{60\text{ sec}} \\ &=& 26.2 \frac{\text{rad}}{\text{s}}\end{array}\\$

Entering this quantity into the expression for $\begin{array}{lll}\alpha & =& \frac{\Delta \omega }{\Delta t}\\ & =& \frac{\text{26.2 rad/s}}{\text{5.00 s}}\\ & =& \text{5.24}{\text{ rad/s}}^{2}\text{.}\end{array}\\$

$\Delta t=\frac{\Delta \omega }{\alpha}\\$

.
$\begin{array}{lll}\Delta t& =& \frac{-\text{26.2 rad/s}}{-\text{87.3}{\text{rad/s}}^{2}}\\ & =& \text{0.300 s.}\end{array}\\$

${a}_{\text{t}}=\frac{\Delta v}{\Delta t}\\$

.
For circular motion, note that ${a}_{\text{t}}=\frac{\Delta \left(\mathrm{r\omega }\right)}{\Delta t}\\$

.
The radius ${a}_{\text{t}}=r\frac{\Delta \omega }{\Delta t}\\$

.
By definition, $\alpha =\frac{\Delta \omega }{\Delta t}\\$

. Thus,$\alpha =\frac{{a}_{\text{t}}}{r}\\$

These equations mean that linear acceleration and angular acceleration are directly proportional. The greater the angular acceleration is, the larger the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car’s drive wheels, the greater the acceleration of the car. The radius also matters. For example, the smaller a wheel, the smaller its linear acceleration for a given angular acceleration
A powerful motorcycle can accelerate from 0 to 30.0 m/s (about 108 km/h) in 4.20 s. What is the angular acceleration of its 0.320-m-radius wheels? (See Figure 4.)

**Strategy**

We are given information about the linear velocities of the motorcycle. Thus, we can find its linear acceleration *a*_{t}. Then, the expression

**Solution**

The linear acceleration is

*a*_{t} and *r* into

**Discussion**

Units of radians are dimensionless and appear in any relationship between angular and linear quantities.

$\alpha =\frac{{a}_{\text{t}}}{r}\\$

can be used to find the angular acceleration.$\begin{array}{lll}{a}_{\text{t}}& =& \frac{\Delta v}{\Delta t}\\ & =& \frac{\text{30.0 m/s}}{\text{4.20 s}}\\ & =& \text{7.14}{\text{m/s}}^{2}\end{array}\\$

.
We also know the radius of the wheels. Entering the values for ${a}_{\text{t}}\\$

and $r$

, we get$\begin{array}{lll}\alpha & =& \frac{{a}_{\text{t}}}{r}\\ & =& \frac{\text{7.14}{\text{m/s}}^{2}}{\text{0.320 m}}\\ & =& \text{22.3}{\text{rad/s}}^{2}\end{array}\\$

.
So far, we have defined three rotational quantities—*θ, ω*, and *α*. These quantities are analogous to the translational quantities *x, v*, and *a*. Table 1 displays rotational quantities, the analogous translational quantities, and the relationships between them.

Rotational | Translational | Relationship |
---|---|---|

θ |
x |
$\theta =\frac{x}{r}\\$ |

ω |
v |
$\omega =\frac{v}{r}\\$ |

α |
a |
$\alpha =\frac{{a}_{t}}{r}\\$ |

Sit down with your feet on the ground on a chair that rotates. Lift one of your legs such that it is unbent (straightened out). Using the other leg, begin to rotate yourself by pushing on the ground. Stop using your leg to push the ground but allow the chair to rotate. From the origin where you began, sketch the angle, angular velocity, and angular acceleration of your leg as a function of time in the form of three separate graphs. Estimate the magnitudes of these quantities.

Angular acceleration is a vector, having both magnitude and direction. How do we denote its magnitude and direction? Illustrate with an example.

Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs.

- Uniform circular motion is the motion with a constant angular velocity $\omega =\frac{\Delta \theta }{\Delta t}\\$.
- In non-uniform circular motion, the velocity changes with time and the rate of change of angular velocity (i.e. angular acceleration) is $\alpha =\frac{\Delta \omega }{\Delta t}\\$.
- Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction, given as ${a}_{\text{t}}=\frac{\Delta v}{\Delta t}\\$.
- For circular motion, note that $v=\mathrm{r\omega }$, so that

${a}_{\mathrm{\text{t}}}=\frac{\Delta \left(\mathrm{r\omega }\right)}{\Delta t}\\$. - The radius r is constant for circular motion, and so $\mathrm{\Delta }\left(\mathrm{r\omega }\right)=r\Delta \omega\\$. Thus,

${a}_{\text{t}}=r\frac{\Delta \omega }{\Delta t}\\$. - By definition, $\Delta \omega /\Delta t=\alpha\\$. Thus,

or${a}_{\text{t}}=\mathrm{r\alpha }\\$

$\alpha =\frac{{a}_{\text{t}}}{r}\\$.

1. Analogies exist between rotational and translational physical quantities. Identify the rotational term analogous to each of the following: acceleration, force, mass, work, translational kinetic energy, linear momentum, impulse.

2. Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude.

3. In circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction. Explain your answer.

4. Suppose a piece of food is on the edge of a rotating microwave oven plate. Does it experience nonzero tangential acceleration, centripetal acceleration, or both when: (a) The plate starts to spin? (b) The plate rotates at constant angular velocity? (c) The plate slows to a halt?

1. At its peak, a tornado is 60.0 m in diameter and carries 500 km/h winds. What is its angular velocity in revolutions per second?

2.** Integrated Concepts **An ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a) What is its angular acceleration in rad/s^{2}? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the radial acceleration in m/s^{2 }and multiples of *g* of this point at full rpm?

3.** Integrated Concepts **You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?

4.** Unreasonable Results **You are told that a basketball player spins the ball with an angular acceleration of 100 rad/s^{2}. (a) What is the ball’s final angular velocity if the ball starts from rest and the acceleration lasts 2.00 s? (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?

- angular acceleration:
- the rate of change of angular velocity with time

- change in angular velocity:
- the difference between final and initial values of angular velocity

- tangential acceleration
- the acceleration in a direction tangent to the circle at the point of interest in circular motion