Chapter 7 Rotational Motion and the Law of Gravity

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Chapter 7 Rotational Motion and the Law of
Gravity
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71 Measuring Rotational Quantities
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Rotational Motion

• Motion of a body that spins about an axis.

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Axis of rotation

• Is the line about which the rotation occurs.

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Angles can be measured in radians (rads).

• Radians an angle whose arc length is equal to
  its radius, which approximately equal to 57.3
  degrees.

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1 radian 57.3o

• s
• ----
• r

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1 radian 57.3o

• s
• ----
• r

s arc length r length of radius q angle of
rotation
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• The angle of 360o is one revolution. (1 rev
  360o)
• One revolution is equal to the circumference of
  the circle of rotation.

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Circumference is 2prso,1 rev 360o 2p rads

• Radius is half the diameter (d).

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Angular Displacement (Dq)

• The angle through which a point, line, or body is
  rotated in a specified direction and about a
  specified axis.

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Describes how far an object has rotated.

• Change
• Angular in arc length
• Displacement --------------
• radius

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• Counterclockwise (CCW) rotation is considered ()
• Clockwise (CW) rotation is considered (-)

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• Ex 1 John Glenn, in 1962, circled the earth 3
  times in less than 5 hours. If his distance from
  the center of the earth was 6560 km, what arc
  length did he travel through?

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• G Dq 3 rev 6p rads, r 6560km
• U Ds ?
• E Ds Dq r
• S Ds (6p)(6560)
• S Ds 123653 km

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Angular Speed (w)

• The rate at which a body rotates about an axis,
  usually expressed in radians per second.
  (rads/sec)

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• Angular
• Average displacement
• Angular -------------------
• Speed time interval

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• Ex 2 In 1975, an ultra-fast centrifuge attained
  an average angular speed of 2.65 x 104 rads/sec.
  What was the angular displacement after 1.5 sec?

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• G wavg 2.65 x 104 rads/sec, Dt 1.5 sec
• U Dq ?
• E Dq wavg Dt
• S Dq (2.65 x 104)(1.5)
• S Dq 39750 radians

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How fast is the earth rotating?

• The Earth rotates at a moderate angular velocity
  of 7.2921159 10 -5 rads/s

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• Ex 3 A water supply tunnel in NYC is 1.70 x 105
  m long and a radius of 2 m. If an electronic eye
  sensor moves around the perimeter of the tunnel
  at an average angular speed of 5.9 rads/sec. How
  long will it take the beetle to travel the
  equivalent of the length of the tunnel?

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• G wavg 5.9 rads/sec, r 2 m, Ds 1.70 x 105
  m
• U Dt ?
• E Dt Dq / wavg
• We need to find Dq.

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• Dq Ds / r
• Dq 1.70 x 105 m / 2 m
• Dq 85000 rads

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• S Dt 85000 rads
• 5.9 rads/sec

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• S Dt 85000 rads
• 5.9 rads/sec
• S Dt 14406.8 sec

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Angular Acceleration (a)

• The time rate of change of angular speed,
  expressed in rads/sec2 .

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wf - wia -------- Dt
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• wf final angular speed
• wi initial angular speed
• a angular acceleration
• Dt elapsed time

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Equation is rearranged just like the linear
acceleration equation

• ALWAYS multiply by Dt first! Then isolate what
  you are looking for.

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Ex 4 A brake is applied to a flywheel for 5
seconds, reducing the angular velocity of the
wheel uniformly from 2.2 rads/sec to 1.8
rads/sec. What is the angular acceleration?
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G Dt 5 sec, w2 1.8 rads/sec, and w1 2.2
rads/sec

• U a ?
• E w2 - w1
• a --------
• Dt

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• 1.8 rads/sec 2.2 rads/sec
• a ------------------------
• 5 sec
• a - 0.08 rads/sec2

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In comparing angular and linear quantities, they
are similar.
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Looking back to Ch 2 equations

• Dx viDt ½ a Dt2
• Dx ½ (vi vf)Dt
• vf vi aDt
• vf2 vi2 2aDx

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We apply angular notation to these equations and
get

• Dq wiDt ½ a Dt2
• Dq ½ (wi wf)Dt
• wf wi aDt
• wf2 wi2 2aDq

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• Ex 5 A wheel on a bike moves through 11.0 rads
  in 2 seconds. What is the angular acceleration of
  the initial angular speed is 2.0 rads/sec?

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• G Dq 11 rads, Dt 2 sec, wi 2 rads/sec
• U a ?
• E Dq wiDt ½ aDt2
• a 2(Dq - wiDt) / Dt2
• S a 211 (2)(2) /22
• S a 3.5 rads/sec2

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7-2 Tangential and Centripetal Acceleration
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Objects in motion have a tangential speed.

• Tangent a line that lies in a plane of circle
  that intersects at one point.

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Tangent line
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Tangential Speed (vt)

• The instantaneous linear speed of an object
  directed along the tangent to the objects path.

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• r distance from axis
• angular speed
• vt tangential speed

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• Ex 6 The radius of a CD is 0.06 m. If a microbe
  riding on the discs rim has a tangential speed
  of 1.88 m/s, what is the microbes angular speed?

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• G r 0.06 m, vt 1.88 m/s
• U w ?
• E w vt / r
• S w 1.88 m/s / 0.06 m
• S w 31.3 rads/sec

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Tangential Acceleration (at)

• The instantaneous linear acceleration of an
  object directed along the tangent to the objects
  circular path

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• at tangential acceleration
• r distance from axis
• a angular acceleration

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• Ex 7 To crack a whip requires its tip to move
  at supersonic speed. This was achieved with a
  whip 56.24 m. If the tip moved in a circle and it
  accelerated from 6.0 rads/sec to 6.3 rads/sec in
  0.6 sec. What would the tangential acceleration
  be?

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• G r 56.24 m, Dt 0.6 sec
• wi 6 rads/sec,
• wf 6.3 rads/sec,
• U at ?
• E at ra

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We need to find angular acceleration (a)

• E wf - wi
• a --------
• t

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• S 6.3 - 6.0
• a --------
• 0.6
• S a 0.5 rads/sec2

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• S at (56.24)(0.5)
• S at 28.12 m/s2

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Centripetal Acceleration (ac)

• Acceleration directed toward the center of a
  circular path.

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• r distance from axis (m)
• ac centripetal accel. (m/s2)
• vt tangential speed (m/s)

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• r distance from axis (m)
• ac centripetal accel. (m/s2)
• w angular speed (rads/sec)

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• Ex 8Calculate the orbital radius of the Earth,
  if its tangential speed is 29.7 km/s and the
  centripetal acceleration acting on the Earth is 5
  x 10-3 m/s2?

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• G vt29.7 km/s 29700 m/s
• ac 5 x 10-3 m/s2
• U r ?
• E vt2
• r ----
• ac

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• S 297002
• r ------------
• 5 x 10-3
• S r 1.764 x 10 11m

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Tangential and centripetal accelerations are
perpendicular.
ac
at
atotal
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The total acceleration can be found by using the
Pythagorean Theorem.
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The total acceleration can be found by using the
Pythagorean Theorem.
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If you need to find the angle (q) use the inverse
tangent function (tan-1)
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7-3 Causes of Circular Motion
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If a mass undergoes an acceleration, there must
be a force causing it. (Newtons 2nd Law)
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A force directed toward the center is necessary
for circular motion.
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This force causing circular motion is called
Centripetal Force (Fc)
Fc
r
m
v
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With a centripetal force, an object in motion
will be accelerated and change its direction.
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Work Done by the Centripetal Force

• Since the centripetal force on an object is
  always perpendicular to the objects velocity,
  the centripetal force never does work on the
  object - no energy is transformed.

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Without a centripetal force, an object in motion
continues along a straight-line path.
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Centrifugal Force

• centrifugal force is a fictitious force - it is
  not an interaction between 2 objects, and
  therefore not a real force.Nothing pulls an
  object away from the center of the circle.

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Centrifugal Force

• What is erroneously attributed to centrifugal
  force is actually the action of the objects
  inertia - whatever velocity it has (speed
  direction) it wants to keep.

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From Newtons 2nd Law F ma

• Fc mac

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From Newtons 2nd Law F ma

• Fc mac

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From Newtons 2nd Law F maFc mac
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• Ex 9 Joe attaches a 0.5 kg mass to a 1m rope.
  Joe swings the rope in a horizontal circle above
  his head with a speed of 4 m/s. What force is
  exerted to keep the mass moving in a circle?

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• G m 0.5 kg
• vt 4 m/s
• r 1 m
• U Fc ?
• E Fc m(vt2 / r)

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• S
• Fc .5 kg x(4 m/s)2/(1 m)
• S Fc 8 N

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Gravitational Force

• The mutual force of attraction between particles
  of matter.

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This is a field force that always exists between
any two objects, regardless of their sizes.
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The gravitational force (Fg) depends upon the
distance between the two masses.
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Newtons Law of Universal Gravitation
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• m1 mass 1 (kg)
• m2 mass 2 (kg)
• r distance between center of masses. (m)
• G Constant of Universal Gravitation.

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Units will cancel out with others leaving just
force units of Newtons (N).
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The universal gravitational constant (G) can be
used to find the gravitational force between any
two particles.
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This is what is known as the inverse-square law.

• The force between the two masses decreases as the
  masses move farther apart.

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The acceleration due to gravity, g, decreases
with distance from the earths center as 1/d2.
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As an object moves away from the center of the
Earth, by a distance equivalent to the earths
radius.
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• Ex 10 What is the force of attraction between
  Adam and Jason, gravitational of course. If they
  are 12 m apart and Jason mass is 82 kg and Adam
  is 74 kg.

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• G m1 82 kg, m2 74kg, r 12 m,
• G 6.673x 10-11 n-m2/kg2
• U Fg ?

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• G m1 82 kg, m2 74kg, r 12 m,
• G 6.673x 10-11 n-m2/kg2
• U Fg ?
• E

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• S Fg 2.81 x 10 -9 N

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• Ex 11 The sun has a mass 2 x 10 30 kg and a
  radius of 7 x 10 5 km. What mass must be located
  on the suns surface for gravitational force of
  470 N to exist between the mass and the sun?

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• G m1 2 x 10 30 kg,
• r 7 x 10 5 km7 x 108 m
• G 6.673x 10-11 n-m2/kg2
• Fg 470 N
• U m2 ?

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E
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E
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S m2 1.7 kg
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