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
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?
Circular Motion and Gravitation Section 1 Tangential Speed (v t ) Speed in a direction tangent to the circle Uniform circular motion: v t 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?
Circular Motion and Gravitation Section 1 Centripetal Acceleration (a c ) 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
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?
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
Circular Motion and Gravitation Section 1 Centripetal Acceleration Click below to watch the Visual Concept. Visual Concept
Circular Motion and Gravitation Section 1 Centripetal Force (F c ) F c ma c and so a F c c v r 2 t mv r 2 t
Circular Motion and Gravitation Section 1 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
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.
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
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?
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?
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.
Circular Motion and Gravitation Section 2 Law of Universal Gravitation F g is proportional to the product of the masses (m 1 m 2 ). F g 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
Circular Motion and Gravitation Section 2 Law of Universal Gravitation
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 (F g ). He solved Newton s equation for G and substituted his experimental values.
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.
Circular Motion and Gravitation Section 2 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?
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.
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.
Circular Motion and Gravitation Section 2 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 a g (9.81 m/s 2 ) The value for g (on Earth) can be calculated as shown below. g F Gmm Gm m mr r g E E 2 2
Circular Motion and Gravitation Section 2 Classroom Practice Problems Find the gravitational force that Earth (m E = 5.97 10 24 kg) exerts on the moon (m m = 7.35 10 22 kg) when the distance between them is 3.84 x 10 8 m. Answer: 1.99 x 10 20 N Find the strength of the gravitational field at a point 3.84 x 10 8 m from the center of Earth. Answer: 0.00270 N/m or 0.00270 m/s 2
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?
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.
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?
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
Circular Motion and Gravitation Section 3 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.
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
Circular Motion and Gravitation Section 3 Equations for Planetary Motion Using SI units, prove that the units are consistent for each equation shown above.
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 10 10 m and its period is 105.5 days. Calculate the mass of the star. Answer: 3.00 10 30 kg What is the velocity of this planet as it orbits the star? Answer: 5.17 10 4 m/s
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.
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.
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.
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.
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?
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.
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 tire 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.
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).
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
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.
Circular Motion and Gravitation Section 4 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.
Circular Motion and Gravitation Section 4 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
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 F out > F in then MA > 1 MA can also be determined from the distances the input and output forces move. MA F out F in d in d out
Circular Motion and Gravitation Section 4 Overview of Simple Machines Click below to watch the Visual Concept. Visual Concept
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.
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. eff W out W in
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.