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Module 7 Rotational Mechanics

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Module Study Objectives. Circular motion (revisited) Law of gravity. Torque ... motion ... torque for a mass in circular motion about a point leads to an ... – PowerPoint PPT presentation

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Title: Module 7 Rotational Mechanics

1
Module 7 Rotational Mechanics
• Serway Faughn Chapters 7 8

2
Module Study Objectives
• Circular motion (revisited)
• Law of gravity
• Torque
• Centre of gravity
• Rotational kinetic energy
• Angular momentum

3
Angular Measure
4
Angular Speed
5
Example
• A helicopter rotor has an angular speed of 320
rpm. What is this in radians per second?

6
Angular Acceleration
7
Rotation Under Constant Angular Acceleration
8
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?

9
Relations Between Angular and Linear Quantities
10
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?

11
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?

12
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.

13
Example
• A test car moves at 10 m/s around a circular road
of radius 50m. Find centripetal acceleration and
angular speed.

14
Centripetal Forces
• Tension
• Gravity
• Friction

v
m
F
r
15
Problem-solving Strategy Centripetal Forces
• Draw a diagram
• Choose coordinate system
• Find the net force towards the centre
• Solve using Fma

16
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?

17
Rotating Systems
• Centrifugal force is based on an erroneous
understanding of motion in an accelerated
reference frame. Objects are merely obeying
Newtons first law.

18
Example
• What speed must a roller coaster car have at the
bottom of a loop of radius 10m to reach the top?

19
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

20
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.

21
Dark Matter
• Star and galaxy motion indicate much more
gravitating matter than is visible.

Orbiting galaxy
Massive galaxy dark matter halo
22
Example
• Use the law of gravity to estimate the Earths
mass.

23
Gravitational Potential Energy
• Newtons law provides an exact expression for
potential energy.
• mgh is a good approximation near the earths
surface.

24
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

25
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.

26
Torque
• The tendency of a force to rotate a body about
some axis is measured by the quantity called the
torque.

d
hinge
F
27
Example
• What is the torque produced by a 300-N force
applied at 600 to the door?

d2.0m
600
hinge
F
28
Torque Equilibrium
• A system is in static equilibrium if
• Resultant external force is zero
• Resultant external torque is zero

29
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
30
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

31
Example
• Determine where the second mass should be for
static equilibrium.

500N
350N
x
1.5m
32
Force and Torque
• The torque for a mass in circular motion about a
point leads to an expression similar to that of
force.

33
Moment of Inertia
• Rotational mechanics analogue of mass.
• Need to sum all contributions with respect to a
specific rotational axis.

34
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.

35
Rotational Kinetic Energy
• A body rotating about some axis with an angular
speed has kinetic energy that depends on its
moment of inertia

36
Angular Momentum
• Product of moment of inertia times angular
velocity.

37
Angular Momentum Torque
• Angular momentum is conserved when the net
external torque acting is zero.

38
Problem-solving for Rotational Motion
• Steps analogous to linear motion strategy
• Analogous equations (eg ?? I? instead of ?Fma)

39
Example
• In a supernova explosion, a star collapses from