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Module 7 Rotational Mechanics

- Serway Faughn Chapters 7 8

Module Study Objectives

- Circular motion (revisited)
- Law of gravity
- Torque
- Centre of gravity
- Rotational kinetic energy
- Angular momentum

Angular Measure

Angular Speed

Example

- A helicopter rotor has an angular speed of 320

rpm. What is this in radians per second?

Angular Acceleration

Rotation Under Constant Angular Acceleration

Example

- A bicycle wheel experiences angular acceleration

of 3.5 rad/s2. If the initial angular speed is

2.0 rad/s, through what angle does the wheel

rotate in 2.0s and what is its speed?

Relations Between Angular and Linear Quantities

Example

- A computer floppy discs rotates from rest to 31.4

rad/s in 0.892s. How many rotations does it make

coming up to speed?

Example

- A CD is read with constant linear speed of 1.3

m/s. What angular speeds are needed at radii of

5.0 8.0cm and how long a track is needed for 1

hour of play?

Centripetal Acceleration

- In circular motionthe centripetal acceleration

is directed inward toward the centre of the

circle and has a magnitude given by either v2/r

or r?2.

Example

- A test car moves at 10 m/s around a circular road

of radius 50m. Find centripetal acceleration and

angular speed.

Centripetal Forces

- Tension
- Gravity
- Friction

v

m

F

r

Problem-solving Strategy Centripetal Forces

- Draw a diagram
- Choose coordinate system
- Find the net force towards the centre
- Solve using Fma

Example

- A car travels 13.4 m/s on a level circular turn,

radius 50.0m. What minimum coefficient of static

friction between tyres and road is needed to

prevent sliding?

Rotating Systems

- Centrifugal force is based on an erroneous

understanding of motion in an accelerated

reference frame. Objects are merely obeying

Newtons first law.

Example

- What speed must a roller coaster car have at the

bottom of a loop of radius 10m to reach the top?

Newtons Law of Gravity

- Every particle in the universe attracts every

other particle with a force that is directly

proportional to the product of their masses and

inversely proportional to the square of the

distance between them

Features of Gravitation

- Gravity is a field force independent of the

medium separating bodies. - Force decreases rapidly with distance.
- Proportional to the product of the masses.
- Force exerted by a spherical mass on an outside

particle acts as if all the mass is at the

centre.

Dark Matter

- Star and galaxy motion indicate much more

gravitating matter than is visible.

Orbiting galaxy

Massive galaxy dark matter halo

Example

- Use the law of gravity to estimate the Earths

mass.

Gravitational Potential Energy

- Newtons law provides an exact expression for

potential energy. - mgh is a good approximation near the earths

surface.

Escape Speed

- An object requires an escape speed to leave the

Earth or some other gravitating body - This speed is 11 km/s for any object (eg gas

molecule) to leave the Earth

Keplers Laws

- All planets move in elliptical orbits with the

Sun at one of the focal points. - A line drawn from the Sun to any planet sweeps

out equal areas in equal time intervals. - The square of the orbital period of any planet is

proportional to the cube of the average distance

from the planet to the Sun.

Torque

- The tendency of a force to rotate a body about

some axis is measured by the quantity called the

torque.

d

hinge

F

Example

- What is the torque produced by a 300-N force

applied at 600 to the door?

d2.0m

600

hinge

F

Torque Equilibrium

- A system is in static equilibrium if
- Resultant external force is zero
- Resultant external torque is zero

Centre of Gravity

- The centre of gravity is where all the mass of

the body can be considered to be concentrated.

THIS END UP

CoG

Problem-solving for Objects in Equilibrium

- Draw a diagram
- Show all the force vectors
- Establish a co-ordinate system
- Apply 2nd equilibrium condition (no net torque)
- Apply 1st condition (no net force) solve

simultaneous equations

Example

- Determine where the second mass should be for

static equilibrium.

500N

350N

x

1.5m

Force and Torque

- The torque for a mass in circular motion about a

point leads to an expression similar to that of

force.

Moment of Inertia

- Rotational mechanics analogue of mass.
- Need to sum all contributions with respect to a

specific rotational axis.

Torque Angular Acceleration

- In general, the total torque on a rigid body

rotating about a fixed axis is given by moment of

inertia times angular acceleration.

Rotational Kinetic Energy

- A body rotating about some axis with an angular

speed has kinetic energy that depends on its

moment of inertia

Angular Momentum

- Product of moment of inertia times angular

velocity.

Angular Momentum Torque

- Angular momentum is conserved when the net

external torque acting is zero.

Problem-solving for Rotational Motion

- Steps analogous to linear motion strategy
- Analogous equations (eg ?? I? instead of ?Fma)

Example

- In a supernova explosion, a star collapses from

radius 106 km to a pulsar of radius 10km

(such events do occur). Assume mass and angular

momentum are preserved. If the star rotated about

once a fortnight (106s), how fast does the pulsar

spin?

End of Module

- Any questions?