Chapter 6 - Circular Motion, Orbits, and Gravity

An object moving in a circle at a constant speed experiences an acceleration directed toward the ______ of the circle.

An object moving in a circle at a constant speed experiences an acceleration directed toward the center of the circle. This is uniform circular motion.

An object moving in a circle has an acceleration toward the ______ , so there must be a net force toward the ______ as well.

An object moving in a circle has an acceleration toward the center, so there must be a net force toward the center as well.

What causes centripetal acceleration?

A particle of mass m moving at constant speed v around a circle of radius r must always have a net force of magnitude mv^2/r pointing toward the center of the circle. The net force causes a centripetal acceleration of circular motion.

What is a satellite?

A satellite is s smaller body of mass m orbiting a larger body of mass M. In other words, the square of the period of the orbit is proportional to the cube of the radius of the orbit.

What is gravitational force?

Each object exerts an attractive force on the other equal in mass.

What is Newton's Inverse-Square Law?

Newton proposed that every object in the universe attracts every other object with a force that has the following properties: 1. The force is inversely proportional to the square of the distance between the objects. 2. The force is directly proportional to the product of the masses of the two objects

What is orbit?

Orbit is a closed trajectory around a planet or star, and in the absence of air resistance, an orbiting projectile is in free fall

What does it mean for motion to be periodic?

Periodic means that an object's path will repeat over and over again.

Rather than specify the time for one revolution, we can specify circular motion by its _________, the number of revolutions per second, for which we use the symbol f.

Rather than specify the time for one revolution, we can specify circular motion by its frequency, the number of revolutions per second, for which we use the symbol f. [Example] An object with a period of one-half second completes 2 revolutions each second.

The riders on a carousel feel pushed out. This isn't a real force, though it is often called centrifugal force; it's an ________ force.

The riders on a carousel feel pushed out. This isn't a real force, though it is often called centrifugal force; it's an apparent force.

How do you find the period of a satellite?

The square of the period of the orbit is proportional to the cube of the radius of the orbit.

The time interval it takes an object to go around a circle one time, completing one revolution (abbreviated rev), is called the ______ of the motion.

The time interval it takes an object to go around a circle one time, completing one revolution (abbreviated rev), is called the period of the motion. Period is represented by the symbol T.

What is centripetal acceleration?

There is an acceleration at every point in the motion, with the acceleration vector pointing toward the center of the circle.

True or False: In a walking gait, your body is in circular motion as you pivot on your forward foot.

True: In a walking gait, your body is in circular motion as you pivot on your forward foot.

True or False: In terms of centripetal acceleration, acceleration depends on speed but also on distance from the center of the circle.

True: In terms of centripetal acceleration, acceleration depends on speed but also on distance from the center of the circle.

True or False: Weightlessness does not occur from an absence of weight or an absence of gravity.

True: Weightlessness does not occur from an absence of weight or an absence of gravity. Objects are in freefall so their apparent weight is zero.

What is the inverse-square law?

Two quantities have an inverse-square relationship if y is inversely proportional to the square of x. [Example] Doubling the distance between two masses causes the force between them to decrease by a factor of 4.

EXAMPLE 6.2 SPINNING SOME TUNES An audio CD has a diameter of 120 mm and spins at up to 540 rpm. When a CD is spinning at its maximum rate, how much time is required for one revolution? If a speck of dust rides on the outside edge of the disk, how fast is it moving? What is the acceleration?

[Answer] For 1 revolution, it take 0.11 s. A speck of dust will travel at 3.4 m/s. The dust accelerates at 190 m/s^2

EXAMPLE 6.3 FIND THE PERIOD OF A CARNIVAL RIDE In the Quasar carnival ride, passengers travel in a horizontal 5.0-m-radius circle. For safe operation, the maximum sustained acceleration that riders may experience is 20 m/s^2, approximately twice the free-fall acceleration. What is the period of the ride when it is being operated at the maximum acceleration? How fast are the riders moving when the ride is operated at this period?

[Answer] For 1 revolution, it take 3.1 s. The riders travel at 10 m/s.

CONCEPTUAL EXAMPLE 6.4 FORCES ON A CAR, PART I Engineers design curves on roads to be segments of circles. They also design dips and peaks in roads to be segments of circles with a radius that depends on expected speeds and other factors. A car is moving at a constant speed and goes into a dip in the road. At the very bottom of the dip, is the normal force of the road on the car greater than, less than, or equal to the car's weight?

[Answer] normal force of the road on the car > car's weight

INTEGRATED EXAMPLE 6.16 A HUNTER AND HIS SLING A Stone Age hunter stands on a cliff overlooking a flat plain. He places a 1.0 kg rock in a sling, ties the sling to a 1.0-m-long vine, then swings the rock in a horizontal circle around his head. The plane of the motion is 25 m above the plain below. The tension in the vine increases as the rock goes faster and faster. Suddenly, just as the tension reaches 200 N, the vine snaps. If the rock is moving toward the cliff at this instant, how far out on the plain (from the base of the cliff) will it land?

[Answer] 32 m

EXAMPLE 6.14 FINDING THE SPEED TO ORBIT DEIMOS Mars has two moons, each much smaller than the earth's moon. The smaller of these two bodies, Deimos, isn't quite spherical, but we can model it as a sphere of radius 6.3 km. Its mass is 1.8 * 10^15 kg. At what speed would a projectile move in a very low orbit around Deimos?

[Answer] 4.4 m/s

EXAMPLE 6.15 LOCATING A GEOSTATIONARY SATELLITE Communication satellites appear to "hover" over one point on the earth's equator. A satellite that appears to remain stationary as the earth rotates is said to be in a geostationary orbit. What is the radius of the orbit of such a satellite?

[Answer] 42,300,000 m

CONCEPTUAL EXAMPLE 6.11 VARYING GRAVITATION FORCE The gravitational force between two giant lead spheres is 0.010 N when the centers of the spheres are 20 m apart. What is the distance between their centers when the gravitational force between them is 0.160 N?

[Answer] 5.0 m

EXAMPLE 6.10 ANALYZING THE ULTRACENTRIFUGE An 18-cm-diameter ultracentrifuge produces an extraordinarily large centripetal acceleration of 250,000g, where g is the free-fall acceleration due to gravity. What is its frequency in rpm? What is the apparent weight of a sample with a mass of 0.0030 kg?

[Answer] 50,000 rpm and 7400 N

EXAMPLE 6.13 GRAVITATIONAL FORCE OF THE EARTH ON PERSON What is the magnitude of the gravitational force of the earth on a 60 kg person? The earth has mass 5.98 * 10^24 kg and radius 6.37 * 10^6 m.

[Answer] 590 N

EXAMPLE 6.9 HOW SLOW CAN YOU GO A motorcyclist in the Globe of Death, pictured here, rides in a 2.2-m-radius vertical loop. To keep control of the bike, the rider wants the normal force on his tires at the top of the loop to equal or exceed his and the bike's combined weight. What is the minimum speed at which the rider can take the loop?

[Answer] 6.6 m/s

EXAMPLE 6.12 GRAVITATIONAL FORCE BETWEEN TWO PEOPLE You are seated in your physics class next to another student 0.60 m away. Estimate the magnitude of the gravitational force between you. Assume that you each have a mass of 65 kg.

[Answer] 7.8 x 10^-7 N

STOP TO THINK 6.2 A block on a string spins in a horizontal circle on a frictionless table. Rank in order, from largest to smallest, the tensions A to E acting on the blocks A to E.

[Answer] A < B < C < D < E The tension force provides the centripetal acceleration, and larger acceleration implies larger force. So the question reduces to one about acceleration: Rank the centripetal accelerations for these cases. Equation 6.5 shows that the acceleration is proportional to the radius of the circle and the square of the frequency, and so the acceleration increases steadily as we move from A to E.

STOP TO THINK 6.6 Each year, the moon gets a little bit farther away from the earth, increasing the radius of its orbit. How does this change affect the length of a month? A. A month gets longer. B. A month gets shorter. C. The length of a month stays the same.

[Answer] A. A month gets longer. The length of a month is determined by the period of the moon's orbit. Equation 6.22 shows that as the moon gets farther away, the period of the orbit—and thus the length of a month—increases.

STOP TO THINK 6.5 Rank in order, from largest to smallest, the free-fall accelerations on the surfaces of the following planets

[Answer] B > A > D > C The free-fall acceleration is proportional to the mass, but inversely proportional to the square of the radius.

STOP TO THINK 6.4 A satellite is in a low earth orbit. Which of the following changes would increase the orbital period? A. Increasing the mass of the satellite. B. Increasing the height of the satellite about the surface. C. Increasing the value of g.

[Answer] B. Increasing the height of the satellite about the surface. The period of a satellite doesn't depend on the mass of the satellite. If you increase the height above the surface, you increase the radius of the orbit. This will result in an increased period. Increasing the value of g would cause the period to decrease.

STOP TO THINK 6.3 A car is rolling over the top of a hill at constant speed v. At this instant, A. n > w. B. n < w. C. n = w. D. We can't tell about n without knowing v.

[Answer] B. n < w The car is moving in a circle, so there must be a net force toward the center of the circle. The center of the circle is below the car, so the net force must point downward. This can be true only if w > n. This makes sense; n < w, so the apparent weight is less than the true weight. The riders in the car "feel light"; if you've driven over a rise like this, you know that this is what you feel.

STOP TO THINK 6.1 Rank in order, from largest to smallest, the period of the motion of particles A to D.

[Answer] C > A = D > B Rearranging Equation 6.3 gives T = ( 2(pi) r )/ v . For the cases shown, speed is either v or 2v; the radius of the circular path is r or 2r. Going around a circle of radius r at a speed v takes the same time as going around a circle of radius 2r at a speed 2v. It's twice the distance at twice the speed.

STOP TO THINK A softball pitcher is throwing a pitch. At the instant shown, the ball is moving in a circular arc at a steady speed. At this instant, the acceleration is A. Directed up. B. Directed down. C. Directed left. D. Directed right. E. Zero.

[Answer] D. Directed right. The ball is in uniform circular motion. The acceleration is directed toward the center of the circle, which is to the right at the instant shown.

EXAMPLE 6.6 ANALYZING THE MOTION OF A CART An energetic father places his 20 kg child on a 5.0 kg cart to which a 2.0-m-long rope is attached. He then holds the end of the rope and spins the cart and child around in a circle, keeping the rope parallel to the ground. If the tension in the rope is 100 N, how much time does it take for the cart to make one rotation?

[Answer] It takes 4.4 s to complete 1 rotation

CONCEPTUAL EXAMPLE 6.1 ROUNDING A CORNER A car is turning a tight corner at a constant speed. A top view of the motion is shown in Figure 6.2. The velocity vector for the car points to the east at the instant shown. What is the direction of the acceleration?

[Answer] South, towards the center of the curve.

EXAMPLE 6.7 FINDING THE MAXIMUM SPEED TO TURN A CORNER What is the maximum speed with which a 1500 kg car can make a turn around a curve of radius 20 m on a level (unbanked) road without sliding? (This radius turn is about what you might expect at a major intersection in a city.)

[Answer] The maximum speed the car can travel without skidding is 14 m/s

EXAMPLE 6.8 FINDING SPEED ON A BANKED TURN A curve on a racetrack of radius 70 m is banked at a 15° angle. At what speed can a car take this curve without assistance from friction?

[Answer] The maximum speed the car can travel without the assistance from friction is 14 m/s

CONCEPTUAL EXAMPLE 6.5 FORCES ON A CAR, PART II A car is turning a corner at a constant speed, following a segment of a circle. What force provides the necessary centripetal acceleration?

[Answer] static friction causes the We know that the acceleration is directed toward the center of the circle. What force or forces can we identify that provide this acceleration? The car's tires against the road must be the friction that causes the car to turn.

What is the formula for critical speed for an object in uniform circular motion?

[Example] The speed for which n = 0 is called the critical speed, the slowest speed at which an object can complete the circle.