© 2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque...

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2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque ( ) is affected by the magnitude (F) , direction(Sin θ ) , and point of application of force ( R ) . = R*F(Sin θ) R = Torque Arm = distance between applied force and pivot point R = Torque Arm = distance between applied force and pivot point Pb.10 – Explain the relationships among speed , velocity , acceleration , and force in rotational systems. http://www.physicsclassroom.com/mmedia/circmot/ circmotTOC.html

Transcript of © 2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque...

Page 1: © 2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque (  ) is affected by the magnitude (F), direction(Sin θ),

© 2010 Pearson Education, Inc.

DoDEA – Physics (SPC 501) - StandardsPb.9 – Explain how torque () is affected by the magnitude (F), direction(Sin θ), and point of application of

force (R). = R*F(Sin θ)

R = Torque Arm = distance between applied force and pivot pointR = Torque Arm = distance between applied force and pivot point

Pb.10 – Explain the relationships among speed, velocity, acceleration, and force in rotational systems.

http://www.physicsclassroom.com/mmedia/circmot/circmotTOC.html

Page 2: © 2010 Pearson Education, Inc. DoDEA – Physics (SPC 501) - Standards Pb.9 – Explain how torque (  ) is affected by the magnitude (F), direction(Sin θ),

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Circular MotionCircular motion (rotation) - When an object

turns about an internal axis

- motion of an object in a circle with a constant or uniform speed (velocity???)

- constant change in direction = acceleration

What is ?

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Period - (cycle) – the time it takes to travel one revolution. -- Object repeatedly finds itself back where it started.

Which “dot” (A-D) is traveling the fastest?

AA BB CC DD

Angular Speed (ω) – the rate at which a body rotates around an axis (rotations per minute = rpm)

AA BB CC DD

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Angular Displacement (θ) - the angle through which a point is rotated

SO, which “dot” (A-D) is traveling the fastest?

distance = rate time

time =distance

rate v

T =2 r

v

2

r

Radial Distance (r) – (radius) – distance from the axis (center)

C = 2 π r

AA BB CC DD

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• Circular Motion is characterized by two kinds of speeds:- Tangential (v) (or linear) speed (circumference / t)

- Angular speed (ω) (rotational or circular) (θ / t)

AA BB CC DD

Angular speed (ω) = “Clock hand speed”

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Circular Motion—Tangential SpeedTangential speed (symbol v) - The distance

traveled by a point on the rotating object divided by the time taken to travel that distance

• Points closer to the circumference (outer edge) have a higher tangential speed that points closer to the center.

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Circular Motion – Rotational Speed• Rotational (angular) speed - (symbol ) - is the

number of rotations or revolutions per unit of time• All parts of a rigid merry-go-round or clock hand

turn about the axis of rotation in the same amount of time. – all parts have the same rotational speed

Tangential speed Radial Distance Rotational Speed

= r x

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A ladybug sits halfway between the rotational axis and the outer edge of the turntable . When the turntable has a rotational speed of 20 RPM and the bug has a tangential speed of 2 cm/s, what will be the rotational and tangential speeds of her friend who sits at the outer edge?

A. 1 cm/sB. 2 cm/s C. 4 cm/sD. 8 cm/s

Rotational and Tangential Speed

CHECK YOUR NEIGHBOR

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A ladybug sits halfway between the rotational axis and the outer edge of the turntable . When the turntable has a rotational speed of 20 RPM and the bug has a tangential speed of 2 cm/s, what will be the rotational and tangential speeds of her friend who sits at the outer edge?

A. 1 cm/sB. 2 cm/s C. 4 cm/sD. 8 cm/s

Rotational and Tangential Speed

CHECK YOUR ANSWER

--Rotational speed () of both bugs is the same (20 RPM)--So if the radial distance (r) doubles, tangential speed (v) will also double.--Tangential speed (v) is 2 cm/s 2 = 4 cm/s.

= r x

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Angular Displacement (θ) - the angle through which a point is rotated

Radial Distance (r) (radius) - distance from the central axis (center)

Angular Speed (ω) – the rate at which a body rotates around an axis (rotations per minute = rpm)

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In circular motion, the direction of any “point” is constantly changing. Therefore, the velocity (speed with direction) is also constantly changing.v1 v2 A change in direction

= A change in velocity= Acceleration a = Δv/t

Acceleration only occurs when there is a net force applied to an object a = F/m

In In which direction which direction must the FORCE be applied must the FORCE be applied in order for an object to continue to travel in in order for an object to continue to travel in constant circular motion? constant circular motion?

Direction of Force Direction of Force = = Direction of AccelerationDirection of Acceleration

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Centripetal Acceleration (Ac) – Acceleration directed toward the center of a circular path --Always points toward center of circle. --(Centripetal = center seeking) --Always changing direction!

NOTENOTE: Velocity is always in a : Velocity is always in a straight linestraight line..

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• Any force directed toward a fixed center is called a centripetal force (Fc). The magnitude of the force required to maintain uniform circular motion.

Example: To whirl a tin can at the end of a string, you pull the string toward the center and exert a centripetal force to keep the can moving in a circle.

Acceleration only occurs when there is a net force applied to an object a = F/m

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• Centripetal Force (Fc) Depends upon:

– Mass (m) of object.– Tangential speed (v) of the object.– Radius (r) of the circle.

radius

2mass tangential speedCentripetal force

F ma

Fmv

r

2

Centripetal Force

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Centripetal Force—Example• When a car rounds a

curve, the centripetal force prevents it from skidding off the road.

• If the road is wet, or if the car is going too fast, the centripetal force is insufficient to prevent skidding off the road.

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Suppose you double the speed at which you round a bend in the curve, by what factor must the centripetal force change to prevent you from skidding?

A. DoubleB. Four timesC. HalfD. One-quarter

Centripetal Force

CHECK YOUR NEIGHBOR

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Suppose you double the speed (v) at which you round a bend in the curve, by what factor must the centripetal force (Fc) change to prevent you from skidding?

A. DoubleB. Four times C. HalfD. One-quarter

Centripetal Force

CHECK YOUR ANSWER

Explanation:

Because the term for “tangential speed” is squared, if you double the tangential speed, the centripetal force will be double squared, which is four times. The radius is constant.

l

2

radius

speed tangentialmass forceCentripeta

WARNINGWARNING: If you make a turn at : If you make a turn at double the speed, you are double the speed, you are 4 times4 times more more

likely to skid off of the road.likely to skid off of the road.

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Suppose you take a sharper turn than before and halve the radius, by what factor will the centripetal force need to change to prevent skidding?

A. DoubleB. Four timesC. HalfD. One-quarter

Centripetal Force

CHECK YOUR NEIGHBOR

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l2

radius

speed tangentialmass forceCentripeta

Suppose you take a sharper turn than before and halve the radius; by what factor will the centripetal force need to change to prevent skidding?

A. DoubleB. Four timesC. HalfD. One-quarter

Centripetal ForceCHECK YOUR ANSWER

Because the term for “radius” is in the denominator, if you halve the radius, the centripetal force will double.

WARNINGWARNING: The sharper your turn : The sharper your turn (smaller radius), the MORE likely you are (smaller radius), the MORE likely you are

to skid. The wider your turn (larger to skid. The wider your turn (larger radius), the LESS likely you are to skid.radius), the LESS likely you are to skid.

What applies this centripetal force to the car???

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Direction of Centripetal Force (Fc), Acceleration (Ac) and Velocity (v)

An object will change direction ONLY if there is a NET force

applied to the object. A CONSTANT force applied

towards a central point results in a circular path

Without a centripetal force, an object in

motion continues along a straight-line path.

Newton’s 1st Law

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The cork will ONLY move while experiencing a net force (resulting in an acceleration). If I were to walk at a constant velocity, the cork would stay in the center.

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Direction of Centripetal Force (Fc), Acceleration (Ac) and Velocity (v)

Velocity (v) = straight line

Force (Fc) = toward center

Acceleration (ac) = toward center

Acceleration occurs in the same direction as the applied force

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A change in velocity is due to?

(α)

radius

2mass tangential speedCentripetal force

F ma

Fmv

r

2

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What if the mass decreases?

radius

2mass tangential speedCentripetal force

F ma

Fmv

r

2

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What if the radius decreases?

radius

2mass tangential speedCentripetal force

F ma

Fmv

r

2

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 A tube is been placed upon the table and shaped into a three-quarters circle. A golf ball is pushed into the tube at one end at high speed. The ball rolls through the tube and exits at the opposite end. Describe the path of the golf ball as it exits the tube.

N

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What provides the centripetal force?

a) Tensionb) Gravityc) Frictiond) Normal Force

Centripetal force is NOT a new “force”. It is simply a way of quantifying the magnitude of the force required to maintain a certain speed around a circular path of a certain radius.

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Tension Can Yield a Centripetal Acceleration:

If the person doubles the speed of the airplane, what happens to the tension in the cable?

F = mamv

r

2

Doubling the speed, quadruples the force (i.e. tension) required to keep the plane in uniform circular motion.

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Friction Can Yield a Centripetal Acceleration:

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F = mamv

r

2

Centripetal Force: Question

Smaller radius: larger force required to keep it in uniform circular motion.

A car travels at a constant speed around two curves. Where is the car most likely to skid? Why?

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A. the same as

B. one fourth of

C. half of

D. twice

E. four times

The answer is E. As the velocity increases the

centripetal force required to maintain

the circle increases as the square of the

speed.

Suppose two identical objects go around in horizontal circles of identical diameter but one object goes around the circle twice as fast as the other. The force required to keep the faster object on the circular path is _____ the force required to keep the slower object on the path.

radius

2mass tangential speedCentripetal force

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Suppose two identical objects go around in horizontal circles with the same speed. The diameter of one circle is half of the diameter of the other. The force required to keep the object on the smaller circular path is ___ the force required to keep the object on the larger path.

A. the same as B. one fourth of C. half of D. twice E. four times

The answer is D. The centripetal force needed to maintain the circular motion of an object is inversely proportional to the radius of the circle. Everybody knows that it is harder to navigate a sharp turn than a wide turn.

radius

2mass tangential speedCentripetal force

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Suppose two identical objects go around in horizontal circles of identical diameter and speed but one object has twice the mass of the other. The force required to keep the more massive object on the circular path is

A. the same as

B. one fourth of

C. half of

D. twice

E. four times

Answer: D.The mass is directly proportional to centripetal force.

radius

2mass tangential speedCentripetal force

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Banked CurvesQ: Why exit ramps in highways are banked?

A: To increase the centripetal force for the higher exit speed.

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The Normal Force Can Yield a Centripetal Acceleration:

How many forces are acting on How many forces are acting on the car (assuming no friction)?the car (assuming no friction)?

Engineers have learned to “bank” curves so that cars can safely travel around the curve without relying on friction at all to supply the centripetal acceleration.

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Banked CurvesWhy exit ramps in highways are banked?

FN cos = mg

Fc = FN sin = mv2/r

The steeper the bank (, the less friction is needed (more Force Normal is applied toward the center). This is why/how a race car track can allow cars to travel at such high speeds.

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Gravity Can Yield a Centripetal Acceleration:

Hubble Space Telescopeorbits at an altitude of 598 km(height above Earth’s surface).What is its orbital speed?

F = mamv

r

2

G m M

R

m v

RE

2

2

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The ride starts to spin faster and faster and you FEEL yourself being pushed back against the metal cage. Even when the ride starts to tilt skyward and you are looking

DOWN at the ground, still you feel yourself almost “forced” back so that you do NOT fall to your death. EXPLAIN….

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Centrifugal Force• Although centripetal force is center directed, an

occupant inside a rotating system seems to experience an outward force. This apparent outward force is called centrifugal force.

• Centrifugal means “center-fleeing” or “away from the center.”

(The F-word)

(The F-word)

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Centrifugal Force – A Common Misconception

• It is a common misconception that a centrifugal force pulls outward on an object.

• Example: – If the string breaks, the object

doesn’t move radially outward. – It continues along its tangent

straight-line path—because no force acts on it. (Newton’s first law)

(The F-word)

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An inward net force is required to make a turn in a circle. This inward net force requirement is known as a centripetal force requirement. In the absence of any net force, an object in motion (such as the passenger) continues in motion in a straight line at constant speed. This is Newton's first law of motion. While the car begins to make the turn, the passenger and the seat begin to edge rightward. In a sense, the car is beginning to slide out from under the passenger. Once striking the driver, the passenger can now turn with the car and experience some circle-like motion. There is never any outward force exerted upon the passenger. The passenger is either moving straight ahead in the absence of a force or moving along a circular path in the presence of an inward-directed force.

http://www.physicsclassroom.com/mmedia/circmot/rht.cfm

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This path is an inward force - a centripetal force. That is spelled c-e-n-t-r-i-p-e-t-a-l, with a "p." The other word - centrifugal, with an "f" - will be considered our forbidden F-word. Simply don't use it and please don't believe in it.

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Rotating Reference Frames• Centrifugal force in a rotating reference

frame is a force in its own right – as real as any other force, e.g. gravity.

• Example:– The bug at the bottom of the can experiences

a pull toward the bottom of the can.

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Why we use the f-word• Centrifugal force in a rotating “reference frame”

can be used to examine the simulation of gravity in space stations of the future.

• By spinning the space station, occupants would experience a centrifugal force (simulated gravity) similar to the bug in the can.

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Simulated Gravity

Have a radius of about 1 km (i.e. diameter of 2 km).

Rotate at a speed of about1 revolution per minute.

To simulate an acceleration due to gravity, g, which is 10 m/s2, a space station must:

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Rotational Inertia• An object rotating about an axis tends to

remain rotating about the same axis at the same rotational speed unless interfered with by some external influence.

• The property of an object to resist changes in its rotational state of motion is called rotational inertia (symbol I).

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Rotational InertiaDepends upon

• mass of object.

• distribution of mass around axis of rotation.– The greater the distance

between an object’s mass concentration and the axis, the greater the rotational inertia.

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Rotational Inertia• The greater the rotational inertia, the

harder it is to change its rotational state.– A tightrope walker carries a long pole that has a high

rotational inertia, so it does not easily rotate.– Keeps the tightrope walker stable.

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Rotational InertiaDepends upon the axis

around which it rotates• Easier to rotate pencil

around an axis passing through it.

• Harder to rotate it around vertical axis passing through center.

• Hardest to rotate it around vertical axis passing through the end.

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Rotational InertiaThe rotational inertia depends upon the shape of the object and its rotational axis.

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A hoop and a disk are released from the top of an incline at the same time. Which one will reach the bottom first?

A. HoopB. Disk C. Both togetherD. Not enough information

Rotational Inertia

CHECK YOUR NEIGHBOR

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A hoop and a disk are released from the top of an incline at the same time. Which one will reach the bottom first?A. HoopB. Disk C. Both togetherD. Not enough information

Rotational Inertia

CHECK YOUR ANSWER

Explanation:Hoop has larger rotational inertia, so it will be slower in gaining speed.

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Torque• The tendency of a force to cause rotation

is called torque.

• Torque depends upon three factors:– Magnitude of the force– The direction in which it acts– The point at which it is applied on the object

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Torque lever arm force

Torque• The equation for Torque is

• The lever arm depends upon– where the force is applied.– the direction in which it acts.

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Torque—Example• 1st picture: Lever arm is less than length of handle

because of direction of force.• 2nd picture: Lever arm is equal to length of handle.• 3rd picture: Lever arm is longer than length of

handle.

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Suppose the girl on the left suddenly is handed a bag of apples weighing 50 N. Where should she sit order to balance, assuming the boy does not move?

A. 1 m from pivotB. 1.5 m from pivotC. 2 m from pivotD. 2.5 m from pivot

Rotational Inertia

CHECK YOUR NEIGHBOR

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Suppose the girl on the left suddenly is handed a bag of apples weighing 50 N. Where should she sit in order to balance, assuming the boy does not move? A. 1 m from pivotB. 1.5 m from pivotC. 2 m from pivotD. 2.5 m from pivot

Rotational Inertia

CHECK YOUR ANSWER

Explanation:She should exert same torque as before.Torque lever arm force

3 m 250 N 750 Nm

Torque new lever arm force750 Nm new lever arm 250N New lever arm 750 Nm / 250 N 2.5 m

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Center of Mass and Center of Gravity

• Center of mass is the average position of all the mass that makes up the object.

• Center of gravity (CG) is the average position of weight distribution. – Since weight and mass are proportional,

center of gravity and center of mass usually refer to the same point of an object.

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Center of Mass and Center of Gravity

To determine the center of gravity,– suspend the object from a point and draw a

vertical line from suspension point. – repeat after suspending from another point.

• The center of gravity lies where the two lines intersect.

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Center of Gravity—StabilityThe location of the center of

gravity is important for stability.

• If we draw a line straight down from the center of gravity and it falls inside the base of the object, it is in stable equilibrium; it will balance.

• If it falls outside the base, it is unstable.

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Angular Momentum• The “inertia of rotation” of rotating objects is

called angular momentum.– This is analogous to “inertia of motion”, which was

momentum.momentum.

• Angular momentum rotational inertia angular velocity

– This is analogous to

Linear momentum mass velocity

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Angular momentum mass tangential speed radius

Angular Momentum• For an object that is small compared with the radial

distance to its axis, magnitude of

– This is analogous to magnitude of

Linear momentum mass speed

• Examples:– Whirling ball at the end of a

long string– Planet going around the Sun

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Angular Momentum

• An external net torque is required to change the angular momentum of an object.

• Rotational version of Newton’s first law: – An object or system of objects will maintain

its angular momentum unless acted upon by an external net torque.

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Suppose you are swirling a can around and suddenly decide to pull the rope in halfway; by what factor would the speed of the can change?

A. DoubleB. Four timesC. HalfD. One-quarter

Angular Momentum

CHECK YOUR NEIGHBOR

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Suppose you are swirling a can around and suddenly decide to pull the rope in halfway, by what factor would the speed of the can change?

A. DoubleB. Four timesC. HalfD. One-quarter

Angular MomentumCHECK YOUR ANSWER

Explanation:

Angular Momentum is proportional to radius of the turn.

No external torque acts with inward pull, so angular momentum is conserved. Half radius means speed doubles.

Angular momentum

mass tangential speed radius

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Conservation of Angular Momentum

The law of conservation of angular momentum states:

If no external net torque acts on a rotating system, the angular momentum of that system remains constant.

Analogous to the law of conservation of linear momentum:If no external force acts on a system, the total linear

momentum of that system remains constant.

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Conservation of Angular Momentum

Example:

• When the man pulls the weights inward, his rotational speed increases!

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Suppose by pulling the weights inward, the rotational inertia of the man reduces to half its value. By what factor would his angular velocity change?

A. DoubleB. Three timesC. HalfD. One-quarter

Angular Momentum

CHECK YOUR NEIGHBOR

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Angular momentum

rotational inertia angular velocity

Suppose by pulling the weights in, if the rotational inertia of the man decreases to half of his initial rotational inertia, by what factor would his angular velocity change?

A. DoubleB. Three timesC. HalfD. One-quarter

Angular MomentumCHECK YOUR ANSWER

Explanation:

Angular momentum is proportional to “rotational inertia”.

If you halve the rotational inertia, to keep the angular momentum constant, the angular velocity would double.

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Vertical Circular Motion