Rotational Motion Describing and Dynamics. Rotational Motion Describing Rotational Motion ...

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Rotational Motion Describing and Dynamics

Transcript of Rotational Motion Describing and Dynamics. Rotational Motion Describing Rotational Motion ...

Page 1: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion

Describing and Dynamics

Page 2: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Describing Rotational Motion

Fractions of revolution measured in grads, degrees, or radians

Grad = revolution

Degree = revolution

Radian = revolution

Page 3: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Angular Displacement

Theta, θ, represents angle of revolution

Counterclockwise rotation positive, clockwise negative

Change in angle = angular displacement

d = rθ

Displacement (d) = rotation through angle,

θ, at distance, r, from center

Rotational Motion

Page 4: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Angular Velocity

Velocity = displacement divided by time

Angular velocity is angular displacement divided by time required to make displacement

Angular velocity is represented by the Greek letter omega, ω

Page 5: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Angular Velocity

If velocity changes over time, average velocity not equal to instantaneous velocity at any given instant

Angular velocity = average angular velocity over a time interval, t

Instantaneous angular velocity = slope of graph of angular position versus time

Measured in rad/s

So, for Earth, ωE = (2π rad)/[(24.0 h)(3600 s/h)] = 7.27×10─5 rad/s

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Rotational Motion Angular Velocity

Counterclockwise rotation also results in positive angular velocity

If angular velocity is ω, then linear velocity of point at distance, r, from axis of rotation given by

Speed at which object on Earth’s equator moves due to Earth’s rotation is v = r ω = (6.38×106 m) (7.27×10─5 rad/s) = 464 m/s

v = rω

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Rotational Motion Angular Acceleration

Change in angular velocity divided by time required to make that change

Measured in rad/s2

If positive, then ω also positive△𝑣

Angular acceleration also average angular acceleration over time interval Δt

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Rotational Motion Angular Acceleration

Find instantaneous angular acceleration by finding slope of graph of angular velocity as function of time

Linear acceleration of point at r from axis of object with angular acceleration, α, given by

a = r

Page 9: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Angular Acceleration

A summary of linear and angular relationships

Page 10: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Angular Frequency

Number of complete revolutions made by object in 1s called angular frequency

Angular frequency, f, is given by the equation:

Page 11: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Describing Rotational Motion

Jupiter, the largest planet in our solar system, rotates around its own axis in 9.84 h. The diameter of Jupiter is 1.43 x 108 m.

What is the angular speed of Jupiter’s rotation in rad/s?

What is the linear speed of a point on Jupiter’s equator, due to Jupiter’s rotation?

A computer disk drive optimizes the data transfer rate by rotating the disk at a constant angular speed of 34.1 rad/s while being read. When the computer is searching for a file, the disk spins for 0.892 s.

What is the angular displacement of the disk during this time?

Through how many revolutions does the disk turn during this time?

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Rotational Motion Describing Rotational Motion

A cyclist wants to complete 10.0 laps around a circular track 1.0 km in diameter in exactly 1.0 h. At what linear velocity must this cyclist ride?

A 75.0 g mass is attached to a 1.0 m length of string and whirled around in the air at a rate of 4.0 rev/s when the string breaks.

What is the breaking force of the string?

What was the linear velocity of the mass as soon as the string broke?

Page 13: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Rotational Dynamics

Change in angular velocity depends on magnitude of force, distance from axis to point where force exerted, and direction of force

To open door, you exert force. Doorknob near outer edge of door. Exert force on doorknob at right angles to door, away from hinges

To get most effect from least force,

exert force as far from axis of

rotation as possible

Page 14: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Rotational Dynamics

Magnitude of force, distance from axis to point where force exerted, and direction of force determine change in angular velocity

Change in angular velocity depends on lever arm, perpendicular distance from axis of rotation to point where force exerted

Page 15: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Rotational Dynamics

For door, distance from the hinges to point where force exerted

If force perpendicular to radius of rotation then lever arm is distance from axis, r. If force not exerted perpendicular to radius, lever arm reduced

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Rotational Motion Rotational Dynamics

Lever arm, L, calculated by equation, L = r sin θ,

θ = angle between force and radius from axis of rotation to point where force applied

Torque measures how effectively force causes

rotation. Measured in newton-meters (N·m)

Page 17: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Rotational Dynamics

Page 18: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Rotational Dynamics

In order for a bolt to be tightened, a torque of 45.0 N•m is needed. You use a 0.341 m long wrench, and you exert a maximum force of 189 N. What is the smallest angle, with respect to the wrench, at which you can exert this force and still tighten the bolt?

Chloe, whose mass is 56 kg, sits 1.2 m from the center of a seesaw. Josh, whose mass is 84 kg, wants to balance Chloe. Where on the seesaw should Josh sit?

Page 19: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion The Moment of Inertia

Equal to mass of object times square

of object’s distance from axis of rotation

Resistance to changes in rotational

motion

Represented by symbol I and has units

of mass times square of distance

Page 20: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Newton’s Second Law for Rotational Motion

Angular acceleration directly proportional to net torque and inversely proportional to moment of inertia

Changes in torque, or moment of inertia, affect rate of rotation

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Rotational Motion Newton’s Second Law for Rotational Motion

A fisherman starts his outboard motor by pulling on a rope wrapped around the outer rim of a flywheel. The flywheel is a solid cylinder with a mass of 9.5 kg and a diameter of 15 cm. The flywheel starts from rest and after 12 s, it rotates at 51 rad/s.

What torque does the fisherman apply to the flywheel (α= τ/I)?

How much force does the fisherman need to exert on the rope to apply this torque?

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Rotational Motion The Center of Mass

The center of mass of an object is the point on the object that moves in the same way that a point particle would move

The path of the center of mass of the object below is a straight line

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Rotational Motion The Center of Mass

To locate the center of mass of an object, suspend the object from any point

When the object stops swinging, the center of mass is along the vertical line drawn from the suspension point

Draw the line, and then suspend the object from another point. Again, the center of mass must be below this point

Draw a second vertical line. The center of mass is at the point where the two lines cross

A wrench, racket, and all other freely-rotating objects, rotate about an axis that goes through their center of mass

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Rotational Motion The Center of Mass

Page 25: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion The Center of Mass of a Human Body

The center of mass of a person varies with posture

For a person standing with his or her arms hanging straight down, the center of mass is a few centimeters below the navel, midway between the front and back of the person’s body

Page 26: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion The Center of Mass of a Human Body

When the arms are raised, as in ballet, the center of mass rises by 6 to10 cm

By raising her arms and legs while in the air, as shown below, a ballet dancer moves her center of mass closer to her head

The path of the center of

mass is a parabola, so the

dancer’s head stays at

almost the same height

for a surprisingly long

time

Page 27: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Center of Mass and Stability

Page 28: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Center of Mass and Stability

An object is said to be stable if an external force is required to tip it

The object is stable as long as the direction of the torque due to its weight, τw tends to keep it upright. This occurs as long as the object’s center of mass lies above its base

To tip the object over, you must rotate its center of mass around the axis of rotation until it is no longer above the base of the object

To rotate the object, you must lift its center of mass. The broader the base, the more stable the object is

Page 29: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Center of Mass and Stability

If the center of mass is outside the base of an object, it is unstable and will roll over without additional torque

If the center of mass is above the base of the object, it is stable

If the base of the object is very narrow and the center of mass is high, then the object is stable, but the slightest force will cause it to tip over

Page 30: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Conditions for Equilibrium

An object is said to be in static equilibrium if both its velocity and angular velocity are zero or constant

First, it must be in translational equilibrium; that is, the net force exerted on the object must be zero

Second, it must be in rotational equilibrium; that is, the net torque exerted on the object must be zero

Page 31: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Rotating Frames of Reference

Newton’s laws are valid only in inertial or nonaccelerated frames

Newton’s laws would not apply in rotating frames of reference, as they are accelerated frames

Motion in a rotating reference frame is important to us because Earth rotates

The effects of the rotation of Earth are too small to be noticed in the classroom or lab, but they are significant influences on the motion of the atmosphere and therefore on climate and weather

Page 32: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Centrifugal “Force”

An observer on a rotating frame, sees an object attached to a spring on the platform

He thinks that some force toward the outside of the platform is pulling on the object

Centrifugal “force” is an apparent force that seems to be acting on an object when that object is kept on a rotating platform

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Rotational Motion Centrifugal “Force”

As the platform rotates, an observer on the ground sees things differently

This observer sees the object moving in a circle

The object accelerates toward the center because of the force of the spring

The acceleration is centripetal acceleration and is given by

Page 34: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion Centrifugal “Force”

It also can be written in terms of angular velocity, as:

Centripetal acceleration is proportional to the distance from the axis of rotation and depends on the square of the angular velocity

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Rotational Motion The Coriolis “Force”

A person standing at the center of a rotating disk throws a ball toward the edge of the disk. An observer standing outside the disk sees the ball travel in a straight line at a constant speed toward the edge of the disk

Page 36: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion The Coriolis “Force”

An observer stationed on the disk and rotating with it sees the ball follow a curved path at a constant speed

A force seems to be acting to deflect the ball

Page 37: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion The Coriolis “Force”

An apparent force that seems to cause deflection to an object in a horizontal motion when the observer is in a rotating frame of reference is known as the Coriolis “force”

It seems to exist because we observe a deflection in horizontal motion when we are in a rotating frame of reference

Page 38: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion The Coriolis “Force”

An observer on Earth, sees the Coriolis “force” cause a projectile fired due north to deflect to the right of the intended target

The direction of winds around high- and low-pressure areas results from the Coriolis “force.” Winds flow from areas of high to low pressure

Page 39: Rotational Motion Describing and Dynamics. Rotational Motion  Describing Rotational Motion  Fractions of revolution measured in grads, degrees, or radians.

Rotational Motion The Coriolis “Force”

Due to the Coriolis “force” in the northern hemisphere, winds from the south blow east of low-pressure areas

Winds from the north, however, end up west of low-pressure areas

Therefore, winds rotate counterclockwise around low-pressure areas in the northern hemisphere

In the southern hemisphere however, winds rotate clockwise around low-pressure areas