circular motion and gravitationsection 1 © houghton mifflin harcourt publishing company what do you...

Circular Motion and Gravitation Section 1

© Houghton Mifflin Harcourt Publishing Company

What do you think?

• Consider the following objects moving in circles at constant speeds`:• A ball tied to a string being swung in a circle • The moon as it travels around Earth• A child riding rapidly on a playground merry-go-round• A car traveling around a circular ramp on the highway

• For each example above, answer the following:• What is keeping the object in the circular path?• Are the objects accelerating?



Ok, what did we learn?

• An object moving in a circle is accelerating.• So, there must be a force.• The force is always pointed towards the center!• This “center- seeking” force is called a

centripetal force centripetal force (LEARN THIS!!!)• The feeling that you are being pulled outward is

your INERTIA and is called centrifugal (center fleeing) and is a FALSE FORCE!!!

Section 1Circular Motion and Gravitation


Tangential Speed (vt)

• Speed in a direction tangent to the circle - AKA linear speed

• Uniform circular motion: vt has a constant value– Only the direction changes

• Angular speed Angular speed is 22/T/T where T is the period.= = / /t = 2t = 2/T/T

• How would the angular speed of a horse near the center of a carousel compare to one near the edge? Tangential? Why?



Tangential Speed (vt)

• (2(2r)/ Tr)/ T• Tangential speed = the

circumference divided by the Period (T)

• The Period is the time of one revolution.



Centripetal Acceleration (ac)

• Acceleration is a change in velocity (speed and/or direction).

• Direction of velocity changes continuously for uniform circular motion.

• What direction is the acceleration?– the same direction as v– toward the center of the circle

• Centripetal means “center seeking”



Centripetal Acceleration (magnitude)

• How do you think the magnitude of the acceleration depends on the speed?

• How do you think the magnitude of the acceleration depends on the radius of the circle?



Tangential Acceleration

• Occurs if the speed CHANGES• Directed tangent to the circle• Example: a car traveling in a circle

– Centripetal acceleration maintains the circular motion.• directed toward center of circle

– Tangential acceleration produces an increase or decrease in the speed of the car.

• directed tangent to the circle



Click below to watch the Visual Concept.

Visual Concept

Centripetal Acceleration



Centripetal Force (Fc)

c cF ma2

and tc

va

r

2

so tc

mvF

r



Centripetal Force

• Maintains motion in a circle• Can be produced in different

ways, such as– Gravity– A string– Friction

• Which way will an object move if the centripetal force is removed?– In a straight line, as shown on

the right



Describing a Rotating System

• Imagine yourself as a passenger in a car turning quickly to the left, and assume you are free to move without the constraint of a seat belt.– How does it “feel” to you during the turn? – How would you describe the forces acting on you during this

turn?

• There is not a force “away from the center” or “throwing you toward the door.”– Sometimes called “centrifugal force”

• Instead, your inertia causes you to continue in a straight line until the door, which is turning left, hits you.



Classroom Practice Problems

• A 35.0 kg child travels in a circular path with a radius of 2.50 m as she spins around on a playground merry-go-round. She makes one complete revolution every 2.25 s.– What is her speed or tangential velocity? (Hint: Find

the circumference to get the distance traveled.)– What is her centripetal acceleration?– What centripetal force is required?

• `Answers: 6.98 m/s, 19.5 m/s2, 682 N



What do you think?

Imagine an object hanging from a spring scale. The scale measures the force acting on the object. • What is the source of this force? What is pulling or

pushing the object downward?• Could this force be diminished? If so, how?• Would the force change in any way if the object was

placed in a vacuum?• Would the force change in any way if Earth stopped

rotating?



Newton’s Thought Experiment• What happens if you fire a

cannonball horizontally at greater and greater speeds?

• Conclusion: If the speed is just right, the cannonball will go into orbit like the moon, because it falls at the same rate as Earth’s surface curves.

• Therefore, Earth’s gravitational pull extends to the moon.



Law of Universal Gravitation

• Fg is proportional to the product of the masses (m1m2).

• Fg is inversely proportional to the distance squared (r2).– Distance is measured center to center.

• G converts units on the right (kg2/m2) into force units (N).– G = 6.673 x 10-11 N•m2/kg2



Law of Universal Gravitation



Gravitational Force

• If gravity is universal and exists between all masses, why isn’t this force easily observed in everyday life? For example, why don’t we feel a force pulling us toward large buildings?– The value for G is so small that, unless at least one of

the masses is very large, the force of gravity is negligible.



Ocean Tides

• What causes the tides?• How often do they occur? • Why do they occur at certain times? • Are they at the same time each day?



Ocean Tides

• Newton’s law of universal gravitation is used to explain the tides. – Since the water directly below the moon is closer than

Earth as a whole, it accelerates more rapidly toward the moon than Earth, and the water rises.

– Similarly, Earth accelerates more rapidly toward the moon than the water on the far side. Earth moves away from the water, leaving a bulge there as well.

– As Earth rotates, each location on Earth passes through the two bulges each day.



Gravity is a Field Force

• Earth, or any other mass, creates a force field.

• Forces are caused by an interaction between the field and the mass of the object in the field.

• The gravitational field (g) points in the direction of the force, as shown.



Calculating the value of g

• Since g is the force acting on a 1 kg object, it has a value of 9.81 N/m (on Earth).– The same value as ag (9.81 m/s2)

• The value for g (on Earth) can be calculated as shown below.

2 2

g E EF Gmm Gm

gm mr r




• Find the gravitational force that Earth

(mE = 5.97 1024 kg) exerts on the moon

(mm= 7.35 1022 kg) when the distance between them is 3.84 x 108 m.– Answer: 1.99 x 1020 N

• Find the strength of the gravitational field at a point 3.84 x 108 m from the center of Earth.– Answer: 0.00270 N/m or 0.00270 m/s2



RPM = rotations per minute T

• If 33 1/3 RPM• Then 1 minute = 33 1/3 rotations• So 60 seconds = 33 1/3 rotations• And 60/33.333 = the period = T



What do you think?

• Electric forces and gravitational forces are both field forces. Two charged particles would feel the effects of both fields. Imagine two electrons attracting each other due to the gravitational force and repelling each other due to the electrostatic force. • Which force is greater?

• Is one slightly greater or much greater than the other, or are they about the same?

• What evidence exists to support your answer?



Coulomb’s Law

• The force between two charged particles depends on the amount of charge and on the distance between them.– Force has a direct relationship with both charges.– Force has an inverse square relationship with distance.



Coulomb’s Law

• Use the known units for q, r, and F to determine the units of kc.

– kc = 9 109 N•m2/C2

• The distance (r) is measured from center to center for spherical charge distributions.



Classroom Practice Problem

• The electron and proton in a hydrogen atom are separated, on the average, a distance of about 5.3 10-11 m. Find the magnitude of both the gravitational force and the electric force acting between them.– Answer: Fe = 8.2 10-8 N, Fg = 3.6 10-47 N

• The electric force is more than 1039 times greater than the gravitational force. – Atoms and molecules are held together by electric

forces. Gravity has little effect.



Compare and contrast



Classroom Practice Problem

• A balloon is rubbed against a small piece of wool and receives a charge of -0.60 C while the wool receives an equal positive charge. Assume the charges are located at a single point on each object and they are 3.0 cm apart. What is the force between the balloon and wool?

• Answer: 3.6 N attractive



Torque

• Where should the cat push on the cat-flap door in order to open it most easily?– The bottom, as far away from the

hinges as possible

• Torque depends on the force (F) and the length of the lever arm (d).



Torque

• Torque also depends on the angle between the force (F) and the distance (d).

• Which situation shown above will produce the most torque on the cat-flap door? Why?– Figure (a), because the force is perpendicular to the distance



Torque

• SI units: N•m– Not joules because torque is not

energy

• The quantity “d ” is the perpendicular distance from the axis to the direction of the force.



Torque as a Vector

• Torque has direction.– Torque is positive if it causes a

counterclockwise rotation.– Torque is negative if it causes a

clockwise rotation.

• Are the torques shown to the right positive or negative?– The wrench produces a positive

torque.– The cat produces a negative

torque.

• Net torque is the sum of the torques.




• Suppose the force on the wrench is 65.0 N and the lever arm is 20.0 cm. Calculate the torque.

• If that was not enough torque to do the job, what could you do?

• Using a cheat pipe, you increase the lever arm to 60.0 cm. What is the torque now?

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