Circular Motion and Gravitation Section 1 Preview Section

Circular Motion and Gravitation Section 1 Preview Section 1 Circular Motion Section 2 Newton’s Law of Universal Gravitation Section 3 Motion in Space Section 4 Torque and Simple Machines © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 1 What do you think? • Consider the following objects moving in circles: • • A car traveling around a circular ramp on the highway 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 • For each example above, answer the following: • Is the circular motion caused by a force? • If so, in what direction is that force acting? • What is the source of the force acting on each object? © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Tangential Speed (vt) • Speed in a direction tangent to the circle • Uniform circular motion: vt has a constant value – Only the direction changes – Example shown to the right • How would the tangential speed of a horse near the center of a carousel compare to one near the edge? Why? © Houghton Mifflin Harcourt Publishing Company Section 1

Circular Motion and Gravitation Centripetal Acceleration (ac) • Acceleration is a change in velocity (size 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” © Houghton Mifflin Harcourt Publishing Company Section 1

Circular Motion and Gravitation Section 1 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? © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 1 Tangential Acceleration • Occurs if the speed increases • 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 © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Centripetal Acceleration Click below to watch the Visual Concept © Houghton Mifflin Harcourt Publishing Company Section 1

Circular Motion and Gravitation Centripetal Force (Fc) © Houghton Mifflin Harcourt Publishing Company Section 1

Circular Motion and Gravitation 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 © Houghton Mifflin Harcourt Publishing Company Section 1

Circular Motion and Gravitation Section 1 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. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 1 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/s 2, 682 N © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 1 Now what do you think? • Consider the following objects moving in circles: • • A car traveling around a circular ramp on the highway 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 • For each example above, answer the following: • Is the circular motion caused by a force? • If so, in what direction is that force acting? • What is the source of the force acting on each object? © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 2 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? © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 2 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. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 2 Law of Universal Gravitation • Fg is proportional to the product of the masses (m 1 m 2). • Fg is inversely proportional to the distance squared (r 2). – Distance is measured center to center. • G converts units on the right (kg 2/m 2) into force units (N). – G = 6. 673 x 10 -11 N • m 2/kg 2 © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Law of Universal Gravitation © Houghton Mifflin Harcourt Publishing Company Section 2

Circular Motion and Gravitation Section 2 The Cavendish Experiment • Cavendish found the value for G. – He used an apparatus similar to that shown above. – He measured the masses of the spheres (m 1 and m 2), the distance between the spheres (r), and the force of attraction (Fg). • He solved Newton’s equation for G and substituted his experimental values. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 2 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. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation 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? © Houghton Mifflin Harcourt Publishing Company Section 2

Circular Motion and Gravitation Section 2 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. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 2 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. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation 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/s 2) • The value for g (on Earth) can be calculated as shown below. © Houghton Mifflin Harcourt Publishing Company Section 2

Circular Motion and Gravitation Section 2 Classroom Practice Problems • Find the gravitational force that Earth (m. E = 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/s 2 © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 2 Now 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? © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 What do you think? • Make a sketch showing the path of Earth as it orbits the sun. • Describe the motion of Earth as it follows this path. • Describe the similarities and differences between the path and motion of Earth and that of other planets. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 What do you think? • What does the term weightless mean to you? • Have you ever observed someone in a weightless environment? If so, when? • How did their weightless environment differ from a normal environment? © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 Kepler’s Laws • Johannes Kepler built his ideas on planetary motion using the work of others before him. – Nicolaus Copernicus and Tycho Brahe © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Kepler’s Laws • Kepler’s first law – Orbits are elliptical, not circular. – Some orbits are only slightly elliptical. • Kepler’s second law – Equal areas are swept out in equal time intervals. © Houghton Mifflin Harcourt Publishing Company Section 3

Circular Motion and Gravitation Section 3 Kepler’s Laws • Kepler’s third law – Relates orbital period (T) to distance from the sun (r) • Period is the time required for one revolution. – As distance increases, the period increases. • Not a direct proportion • T 2/r 3 has the same value for any object orbiting the sun © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 Equations for Planetary Motion • Using SI units, prove that the units are consistent for each equation shown above. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 Classroom Practice Problems • A large planet orbiting a distant star is discovered. The planet’s orbit is nearly circular and close to the star. The orbital distance is 7. 50 1010 m and its period is 105. 5 days. Calculate the mass of the star. – Answer: 3. 00 1030 kg • What is the velocity of this planet as it orbits the star? – Answer: 5. 17 104 m/s © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 Weight and Weightlessness • Bathroom scale – A scale measures the downward force exerted on it. – Readings change if someone pushes down or lifts up on you. • Your scale reads the normal force acting on you. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 Apparent Weightlessness • Elevator at rest: the scale reads the weight (600 N). • Elevator accelerates downward: the scale reads less. • Elevator in free fall: the scale reads zero because it no longer needs to support the weight. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 Apparent Weightlessness • You are falling at the same rate as your surroundings. – No support force from the floor is needed. • Astronauts are in orbit, so they fall at the same rate as their capsule. • True weightlessness only occurs at great distances from any masses. – Even then, there is a weak gravitational force. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 Now what do you think? • Make a sketch showing the path of Earth as it orbits the sun. • Describe the motion of Earth as it follows this path. • Describe the similarities and differences between the path and motion of Earth and that of other planets. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 3 Now what do you think? • What does the term weightless mean to you? • Have you ever observed someone in a weightless environment? If so, when? • How did their weightless environment differ from a normal environment? © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 4 What do you think? • Doorknobs come in a variety of styles. Describe some that you have seen. • Which style of doorknob is easiest to use? Why? • List the names of any simple machines you can recall. • What is the purpose of a simple machine? • Provide an example. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 4 Rotational and Translational Motion • Consider a tire on a moving car. – Translational motion is the movement of the center of mass. • The entire is changing positions. – Rotational motion is the movement around an axis. • Rotation occurs around a center. • Changes in rotational motion are caused by torques. – Torque is the ability of a force to affect rotation. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 4 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). © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 4 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 © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 4 Torque • SI units: N • m – Not joules because torque is not energy • The quantity “d sin ” is the perpendicular distance from the axis to the direction of the force. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation 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. © Houghton Mifflin Harcourt Publishing Company Section 4

Circular Motion and Gravitation Classroom Practice Problems • Suppose the force on the wrench is 65. 0 N and the lever arm is 20. 0 cm. The angle ( ) between the force and lever arm is 35. 0°. Calculate the torque. – Answer: 7. 46 N • m • What force would be required to produce the same torque if the force was perpendicular to the lever arm? – Answer: 37. 3 N © Houghton Mifflin Harcourt Publishing Company Section 4

Circular Motion and Gravitation Section 4 Simple Machines • Change the size or direction of the input force • Mechanical advantage (MA) compares the input force to the output force. – When Fout > Fin then MA > 1 • MA can also be determined from the distances the input and output forces move. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Overview of Simple Machines Click below to watch the Visual Concept © Houghton Mifflin Harcourt Publishing Company Section 4

Circular Motion and Gravitation Section 4 Simple Machines • Simple machines alter the force and the distance moved. • For the inclined plane shown: – F 2 < F 1 so MA >1 and d 2 > d 1 • If the ramp is frictionless, the work is the same in both cases. – F 1 d 1 = F 2 d 2 • With friction, F 2 d 2 > F 1 d 1. – The force is reduced but the work done is greater. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 4 Efficiency of Simple Machines • Efficiency measures work output compared to work input. – In the absence of friction, they are equal. • Real machines always have efficiencies less than 1, but they make work easier by changing the force required to do the work. © Houghton Mifflin Harcourt Publishing Company

Circular Motion and Gravitation Section 4 Now what do you think? • Doorknobs come in a variety of styles. Describe some that you have seen. • Which style of doorknob is easiest to use? Why? • List the names of any simple machines you can recall. • What is the purpose of a simple machine? • Provide an example. © Houghton Mifflin Harcourt Publishing Company