1. Circular Motion

• Our lives are filled with examples of objects moving in circular paths, from carnival rides to our planet orbiting the sun.

• Circular motion is always around a straight line called an axis.

• Rotation is the spinning motion that takes place when an object moves around an axis located within the object.

• Revolution is the circular motion that takes place when an object moves around an axis located outside the object.

2. Examples

• The Earth rotates around its internal axis and revolves around an external axis through the sun.

• Does a tossed football rotate or revolve?

• Does a ball whirled overhead at the end of a string rotate or revolve?

3. Rotational Speed

• For a rotating object we must consider two different measures of speed. rotational or angular speed (). linear or tangential speed (v).

• Rotational speed is the number of rotations per unit time for a body around its axis. rotations / minute degrees / second radians / second

• All parts of a rigid body will rotate about their axis in the same amount of time.

• All parts of the body have the same rotational speed.

4. Tangential Speed

• Tangential speed is the distance moved per unit time by an object along circular path.

• The tangential speed of an object moving in a circular path increases with the radial distance from the axis due to the increase in the circumference.

• Tangential speed radial distance x rotational speed

• v = r x r in meters in rad / s

Rotational and Tangential Speed

constant

t

variest

dv

•Review Questions 1. Distinguish between a rotation and a revolution.• Rotation – axis within object; Revolution – axis outside•2.Does a child on a merry-go-round revolve or rotate …• Revolves•3. Distinguish between linear and rotational speed.• Linear – distance/time; rotational – angle/time• 4. What is linear speed called when something moves in a circle?• Tangential speed.•5. At a given distance from the axis, how does linear speed change as rotational speed changes?• Directly v = r•6. At a given rotational speed, how does linear speed change as the distance from the axis changes.•Directly v = r

•7. When you roll a cylinder across a surface it follows a straight-line path. A tapered cup rolled on the same surface follows a circular path. Why?• Outer diameter has greater linear speed.•Think and Explain 1.If you lose your grip on a rapidly spinning merry-go-round and fall off, in which direction will you fly?•Tangent to edge.•2.A ladybug sits halfway between the axis and the edge of a rotating turntable. What will happen to the ladybug’s linear speed if: the RMP is doubled; the ladybug sits at the edge; both occur?•Doubles; doubles; quadruples. •3.Which state in the US has the greatest linear speed?• Hawaii, closest to equator (r around axis is greatest)

•4. The speedometer in a car is driven by a cable connected to the shaft that turns the car’s wheels. Will speedometer readings be more or less than actual speed when the car’s wheels are replaced with smaller ones.•More•5….A taxi driver wishes to increase his fares by adjusting air pressure in his tires. Should he inflate the tires at low or high pressure?•Low•Think and Solve 1.Mars is about twice the distance from the sun as is Venus. A Martian year is about three times as long as a Venusian year. Which of the two planets has the greater rotational speed? Which planet has the greater linear speed?•Venus v = 3m; Venus vv = vrv = 3m(rm/2) = 3/2vm

•2. Turntable revolves at 10 revolutions / sec. with a laser beam that sweeps across clouds that are 10km away. How fast does the spot sweep across the clouds/• vv = 2r/t = 2(10km)/0.1s=628km/s•How fast would the spot travel if clouds are 20 km away?• v r therefore vv = 1,256km/s•What would r have to be for vv=3,000km/s?

•Practice Page 24c

kmskm

skmkmr 4777)

/628

/000,300(10

1. Circular Motion

• Simulation 21- The Sling

• A change in the direction of velocity is acceleration.

• Circular motion is the movement of an object at a constant speed around a curved or a circular path with a fixed radius.

2. Circular Motion Diagram

Centripetal acceleration

Velocity

V

V

ac

ac

radius

3. Circular Motion

• Magnitude of velocity (speed) does not change.

• Direction of velocity is continually changing.

• Caused by a centripetal (directed toward center) acceleration.

• Velocity and acceleration are perpendicular.

• If the object is released at any point in time it will move with a velocity tangential to the circle.

4. Centripetal Force

• In order to have acceleration (F = ma) an unbalanced force must be present.

• A centripetal force causes an object to follow a curved or circular path.

• A centripetal force acts toward the center of a curved or circular path.

• Examples friction between wheels of car and curved road gravity between planet and sun others

5. Centripetal Force Diagram

Centripetal acceleration

Velocity

V

V

ac

ac

radius

Centripetal Force

•Review Questions 8. When you whirl a can at the end of a string in a circular path, what is the direction of the force that acts on the can?•Toward the center of the circle (your hand)•9.Does the force that holds the riders on the carnival ride in Fig. 9-1 act toward or away from the center?•Toward the center.•10. Does an inward force or an outward force act on the clothes during the spin cycle of an automatic washer?•Inward force•Think and Explain 7.A motorcycle is able to ride on the vertical wall of a bowl-shaped. Does the centripetal force or “centrifugal” force act on the motorcycle?•Centripetal force acts inward.

•8. When a soaring eagle turns during its flight, what is the source of the centripetal force acting on it?•Air against wings•9. Do you think that the road could be banked so that for a given speed and a given radius of curvature a vehicle could make a turn without friction?• Yes, if the horizontal component of the normal force is equal to the necessary centripetal force.•10. Does centripetal force do work on a rotating object?• No, the direction of the force is always perpendicular to the direction of motion.

Physics 8H Announcements

Today • Centripetal Force• Simulated Gravity

Homework• None

Wednesday• Center of Gravity

1. Centripetal Acceleration and Centripetal Force Formulae

• Frog on Turntable Simulation

r

vac

2

cc maF

r

vmFc

2

22)( mr

r

rmFc

2. “Centrifugal” Force is a Misconception

• What would happen if the friction force between the frog’s feet and the turntable is insufficient to hold the frog on the table?

• What force is acting on the frog after it leaves the turntable?

• The frog moves in a tangential straight-line path due to inertia.

• “Centrifugal” force is a fictitious force which is a common misconception.

3. Simulated Gravity

• Within a stationary frame of reference centripetal force produces circular motion.

• Within a frame of reference that is rotating (non-inertial – Newton’s 1st Law does not apply) an observer feels an action-reaction pair of forces which s/he interprets as “gravity”

• This is fictitious force since it is not caused by a planetary body.

• To simulate 1 g at 1 RPM would require a large structure of 2km in diameter.

• The simulated value of g would vary with the radius from the center.

•Review Questions 14. A Ladybug in the bottom of a whirling can feels a “centrifugal” force pushing it against the bottom of the can. Is there an outside source of this force? Can you identify this as the action force of an action-reaction pair? If so, what is the reaction force?•No; no; there is no reaction force.•15. Why is the “centrifugal” force the ladybug feels in the rotating frame called a fictitious force?• No outside source of this force.•16. For a rotating space habitat of a given size, what is the relationship between the magnitude of simulated gravity and the habitat’s rate of spin?•Simulated gravity 2

•17. For space habitats spinning at the same rate, what is the relationship between simulated gravity and the radius of the habitat?• Simulated gravity r•18.Why will orbiting space stations that simulate gravity likely be large structures?•To simulate gravity at low rotational speeds that are not as perceptible.•Think and Explain 12. Occupants in a single space shuttle in orbit feel weightless. Describe a scheme whereby occupants in a pair of shuttles would be able to use a long cable to continuously a comfortably normal earthlike gravity.• Join them together with cable and rotate.

•Think and Solve 3. Consider a too-small space station that consists of a 4m radius rotating sphere. A man standing inside is 2m tall and his feet are at 1 g. What is the acceleration of his head? Explain why rotating structures need to be small.• 0.5g. Large to the difference in simulated gravity across the body is small.•4. Standing inside a rotating space station, your feet have a greater linear speed and a greater centripetal acceleration than your head. We say there is a difference in g’s from your head to your feet, and this difference can be quite uncomfortable. Studies show, however, that a difference of 1/100g produces no discomfort. What should be the radius of the space station compared to your height.•Δg = h/r, r = 100 x height