Chapter 6,9,10

- Circular Motion, Gravitation, Rotation, Bodies in

Equilibrium

Circular Motion

- Ball at the end of a string revolving
- Planets around Sun
- Moon around Earth

The Radian

- The radian is a unit of angular measure
- The radian can be defined as the arc length s

along a circle divided by the radius r

57.3

More About Radians

- Comparing degrees and radians
- Converting from degrees to radians

Angular Displacement

- Axis of rotation is the center of the disk
- Need a fixed reference line
- During time t, the reference line moves through

angle ?

Angular Displacement, cont.

- The angular displacement is defined as the angle

the object rotates through during some time

interval - The unit of angular displacement is the radian
- Each point on the object undergoes the same

angular displacement

Average Angular Speed

- The average angular speed, ?, of a rotating rigid

object is the ratio of the angular displacement

to the time interval

Angular Speed, cont.

- The instantaneous angular speed
- Units of angular speed are radians/sec
- rad/s
- Speed will be positive if ? is increasing

(counterclockwise) - Speed will be negative if ? is decreasing

(clockwise)

Average Angular Acceleration

- The average angular acceleration of an object

is defined as the ratio of the change in the

angular speed to the time it takes for the object

to undergo the change

Angular Acceleration, cont

- Units of angular acceleration are rad/s²
- Positive angular accelerations are in the

counterclockwise direction and negative

accelerations are in the clockwise direction - When a rigid object rotates about a fixed axis,

every portion of the object has the same angular

speed and the same angular acceleration

Angular Acceleration, final

- The sign of the acceleration does not have to be

the same as the sign of the angular speed - The instantaneous angular acceleration

Analogies Between Linear and Rotational Motion

Linear Motion with constant acc. (x,v,a)

Rotational Motion with fixed axis and constant

a (q,?,a)

Examples

- 78 rev/min?
- A fan turns at a rate of 900 rpm
- Tangential speed of tips of 20cm long blades?
- Now the fan is uniformly accelerated to 1200 rpm

in 20 s

Relationship Between Angular and Linear Quantities

- Displacements
- Speeds
- Accelerations

- Every point on the rotating object has the same

angular motion - Every point on the rotating object does not have

the same linear motion

Centripetal Acceleration

- An object traveling in a circle, even though it

moves with a constant speed, will have an

acceleration - The centripetal acceleration is due to the change

in the direction of the velocity

Centripetal Acceleration, cont.

- Centripetal refers to center-seeking
- The direction of the velocity changes
- The acceleration is directed toward the center of

the circle of motion

Centripetal Acceleration, final

- The magnitude of the centripetal acceleration is

given by - This direction is toward the center of the circle

Centripetal Acceleration and Angular Velocity

- The angular velocity and the linear velocity are

related (v ?R) - The centripetal acceleration can also be related

to the angular velocity

Forces Causing Centripetal Acceleration

- Newtons Second Law says that the centripetal

acceleration is accompanied by a force - F ma ?
- F stands for any force that keeps an object

following a circular path - Tension in a string
- Gravity
- Force of friction

Examples

- Ball at the end of revolving string
- Fast car rounding a curve

More on circular Motion

- Length of circumference 2?R
- Period T (time for one complete circle)

Example

- 200 grams mass revolving in uniform circular

motion on an horizontal frictionless surface at 2

revolutions/s. What is the force on the mass by

the string (R20cm)?

Newtons Law of Universal Gravitation

- Every particle in the Universe attracts every

other particle with a force that is directly

proportional to the product of the masses and

inversely proportional to the square of the

distance between them.

Universal Gravitation, 2

- G is the constant of universal gravitational
- G 6.673 x 10-11 N m² /kg²
- This is an example of an inverse square law

Universal Gravitation, 3

- The force that mass 1 exerts on mass 2 is equal

and opposite to the force mass 2 exerts on mass 1 - The forces form a Newtons third law

action-reaction

Universal Gravitation, 4

- The gravitational force exerted by a uniform

sphere on a particle outside the sphere is the

same as the force exerted if the entire mass of

the sphere were concentrated on its center

Gravitation Constant

- Determined experimentally
- Henry Cavendish
- 1798
- The light beam and mirror serve to amplify the

motion

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Applications of Universal Gravitation

- Weighing the Earth

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Applications of Universal Gravitation

- g will vary with altitude

Escape Speed

- The escape speed is the speed needed for an

object to soar off into space and not return - For the earth, vesc is about 11.2 km/s
- Note, v is independent of the mass of the object

Various Escape Speeds

- The escape speeds for various members of the

solar system - Escape speed is one factor that determines a

planets atmosphere

Motion of Satellites

- Consider only circular orbit
- Radius of orbit r
- Gravitational force is the centripetal force.

Motion of Satellites

- Period ?

Keplers 3rd Law

Communications Satellite

- A geosynchronous orbit
- Remains above the same place on the earth
- The period of the satellite will be 24 hr
- r h RE
- Still independent of the mass of the satellite

Satellites and Weightlessness

- weighting an object in an elevator
- Elevator at rest mg
- Elevator accelerates upward m(ga)
- Elevator accelerates downward m(ga) with alt0
- Satellite a-g!!

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Force vs. Torque

- Forces cause accelerations
- Torques cause angular accelerations
- Force and torque are related

Torque

- The door is free to rotate about an axis through

O - There are three factors that determine the

effectiveness of the force in opening the door - The magnitude of the force
- The position of the application of the force
- The angle at which the force is applied

Torque, cont

- Torque, t, is the tendency of a force to rotate

an object about some axis - t is the torque
- F is the force
- symbol is the Greek tau
- l is the length of lever arm
- SI unit is N.m
- Work done by torque W??

Direction of Torque

- If the turning tendency of the force is

counterclockwise, the torque will be positive - If the turning tendency is clockwise, the torque

will be negative

Multiple Torques

- When two or more torques are acting on an object,

the torques are added - If the net torque is zero, the objects rate of

rotation doesnt change

Torque and Equilibrium

- First Condition of Equilibrium
- The net external force must be zero
- This is a necessary, but not sufficient,

condition to ensure that an object is in complete

mechanical equilibrium - This is a statement of translational equilibrium

Torque and Equilibrium, cont

- To ensure mechanical equilibrium, you need to

ensure rotational equilibrium as well as

translational - The Second Condition of Equilibrium states
- The net external torque must be zero

Equilibrium Example

- The woman, mass m, sits on the left end of the

see-saw - The man, mass M, sits where the see-saw will be

balanced - Apply the Second Condition of Equilibrium and

solve for the unknown distance, x

Moment of Inertia

- The angular acceleration is inversely

proportional to the analogy of the mass in a

rotating system - This mass analog is called the moment of inertia,

I, of the object - SI units are kg m2

Newtons Second Law for a Rotating Object

- The angular acceleration is directly proportional

to the net torque - The angular acceleration is inversely

proportional to the moment of inertia of the

object

More About Moment of Inertia

- There is a major difference between moment of

inertia and mass the moment of inertia depends

on the quantity of matter and its distribution in

the rigid object. - The moment of inertia also depends upon the

location of the axis of rotation

Moment of Inertia of a Uniform Ring

- Image the hoop is divided into a number of small

segments, m1 - These segments are equidistant from the axis

Other Moments of Inertia

Example

- Wheel of radius R20 cm and I30kgm². Force

F40N acts along the edge of the wheel. - Angular acceleration?
- Angular speed 4s after starting from rest?
- Number of revolutions for the 4s?
- Work done on the wheel?

Rotational Kinetic Energy

- An object rotating about some axis with an

angular speed, ?, has rotational kinetic energy

KEr½I?2 - Energy concepts can be useful for simplifying the

analysis of rotational motion - Units (rad/s)!!

Total Energy of a System

- Conservation of Mechanical Energy
- Remember, this is for conservative forces, no

dissipative forces such as friction can be

present - Potential energies of any other conservative

forces could be added

Rolling down incline

- Energy conservation
- Linear velocity and angular speed are related

vR? - Smaller I, bigger v, faster!!

Work-Energy in a Rotating System

- In the case where there are dissipative forces

such as friction, use the generalized Work-Energy

Theorem instead of Conservation of Energy - (KEtKERPE)iW(KEtKERPE)f

Angular Momentum

- Similarly to the relationship between force and

momentum in a linear system, we can show the

relationship between torque and angular momentum - Angular momentum is defined as
- L I ?
- and

Angular Momentum, cont

- If the net torque is zero, the angular momentum

remains constant - Conservation of Angular Momentum states The

angular momentum of a system is conserved when

the net external torque acting on the systems is

zero. - That is, when

Conservation Rules, Summary

- In an isolated system, the following quantities

are conserved - Mechanical energy
- Linear momentum
- Angular momentum

Conservation of Angular Momentum, Example

- With hands and feet drawn closer to the body, the

skaters angular speed increases - L is conserved, I decreases, w increases

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Example

- A 500 grams uniform sphere of 7.0 cm radius spins

at 30 rev/s on an axis through its center. - Moment of inertia
- Rotational kinetic energy
- Angular momentum

Example

- Find work done to open 30? a 1m wide door with a

steady force of 0.9N at right angle to the

surface of the door.

Example

- A turntable is a uniform disk of metal of mass

1.5 kg and radius 13 cm. What torque is required

to drive the turntable so that it accelerates at

a constant rate from 0 to 33.3 rpm in 2 seconds?

Center of Gravity

- The force of gravity acting on an object must be

considered - In finding the torque produced by the force of

gravity, all of the weight of the object can be

considered to be concentrated at a single point

Calculating the Center of Gravity

- The object is divided up into a large number of

very small particles of weight (mg) - Each particle will have a set of coordinates

indicating its location (x,y)

Calculating the Center of Gravity, cont.

- We wish to locate the point of application of the

single force whose magnitude is equal to the

weight of the object, and whose effect on the

rotation is the same as all the individual

particles. - This point is called the center of gravity of the

object

Coordinates of the Center of Gravity

- The coordinates of the center of gravity can be

found

Center of Gravity of a Uniform Object

- The center of gravity of a homogenous, symmetric

body must lie on the axis of symmetry. - Often, the center of gravity of such an object is

the geometric center of the object.

Example

- Find the center of mass (gravity) of these

masses 3kg (0,1), 2kg (0,0) - And 1kg (2,0)

Example

- Find the center of mass (gravity) of the

dumbbell, 4 kg and 2 kg with a 4m long 3kg rod.

Torque, review

- t is the torque
- F is the force
- symbol is the Greek tau
- l is the length of lever arm
- SI unit is N.m

Direction of Torque

- If the turning tendency of the force is

counterclockwise, the torque will be positive - If the turning tendency is clockwise, the torque

will be negative

Multiple Torques

- When two or more torques are acting on an object,

the torques are added - If the net torque is zero, the objects rate of

rotation doesnt change

Example

- A 2 m by 2 m square metal plate rotates about its

center. Calculate the torque of all five forces

each with magnitude 50N.

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Torque and Equilibrium

- First Condition of Equilibrium
- The net external force must be zero

- The Second Condition of Equilibrium states
- The net external torque must be zero

Example

- The system is in equilibrium. Calculate W and

find the tension in the rope (T).

Example

- A 160 N boy stands on a 600 N concrete beam in

equilibrium with two end supports. If he stands

one quarter the length from one support, what are

the forces exerted on the beam by the two

supports?

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