Chapter 6

Newton's law of gravity describes the gravitational force between __________.

-the sun and the planets. -Earth and the sun. -a person and the earth. -Earth and the moon.

For a person riding a roller coaster who is upside down at the top of a loop-the-loop, which of the following statements are true? Choose all that apply. For a person riding a roller coaster who is upside down at the top of a loop-the-loop, which of the following statements are true? Choose all that apply. The person's apparent weight is always equal to their actual weight. The normal force from the seat on the person points downward. The magnitude of the normal force from the seat on the person depends on how fast the person is moving. The normal force from the seat on the person points upward.

1) The persons apparent weight is always equal to their apparent weight FALSE apparent weight depends on the speed, but actual weight is constant 2) Normal force from the seat points downwards TRUE As the centrifugal force directed is greater than weight directed downwards, normal reaction is directed downwards 3) The magnitude of normal force depends on how fast the person is moving TRUE Normal reaction is given by N=mv*2/R - mg 4) normal force points upwards FALSE

How many times per day does the International Space Station—or any satellite in a similar low orbit—go around the earth?

15

If a planet has the same mass as the earth, but has twice the radius, how does the surface gravity, g, compare to g on the surface of the earth?

4 times smaller

A coin is rotating on a turntable; it moves without sliding. At the instant shown, suppose the frictional force disappeared. In what direction would the coin move? n-a,s-c, e-b or w-d

A

Dynamics of Uniform Circular Motion

According to Newton's second law, the riders must have a net force acting on them. net force: fnet=ma=mv*2/r, toward center of circle Net force (=ma) producing the centripetal acceleration of uniform circular motion. ***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.

Critical Speed

As the car goes slower there comes a point where n becomes 0: wapp=n=mv*2/t - w vc- bc for n to be zero we must have mvc*2/r=w vc=\/rw/m= \/rmg/m = \/gr The critical speed is the slowest speed at which the car can complete the circle

A ball at the end of a string is being swung in a horizontal circle. The ball is accelerating because A. The speed is changing. B. The direction is changing. C. The speed and the direction are changing. D. The ball is not accelerating.

B. The direction is changing.

A ball at the end of a string is being swung in a horizontal circle. What is the direction of the acceleration of the ball? A. Tangent to the circle, in the direction of the ball's motion B. Toward the center of the circle

B. Toward the center of the circle

A hollow tube lies flat on a table. A ball is shot through the tube. As the ball emerges from the other end, which path does it follow? A, B or C

C - Newton's first law, there is no C force anymore. The object will continue with the same speed. the speed is tangent to the circle, so it will remain so.

Problem-Solving Approach 6.1 Circular Dynamics Problems

Circular motion involves an acceleration and this a net force. Use Newton's 2nd Law problems. Strategy: The acceleration is caused by a net force directed toward the center. The origin of this force is tension, weight, or friction. Prepare: Create a pictorial representation - sketch the motion, define symbols, define axes, ad identify what the problem is trying to find. There are two common situations: Prepare: If the motion is in a horizontal plane, like a tabletop, draw with the circle viewed edge-on, the x-axis pointing toward the center of the cirlce and the y-axis perpendicular to the plane of the circle. OR If the motion is in a vertical plane, like a ferris wheel, draw the free-body diagram with the circle viewed face-on, the x-axis pointing toward the center of the circle, and the y-axis tangent to the circle. Solve: Use Newton's 2nd law for uniform circular motion, Fnet=mv*2/r, toward circle as a vector equation. Solve: -Use the net force to determine the speed v, then use circular kinematics to find frequencies or other details of the motion. -Use circular kinematics to determine the speed v, then solve for unknown forces.

A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the frictional force on the coin? n-a,s-c, e-b or w-d

D

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.

Directed Right

An ice hockey puck is tied by a string to a stake in the ice. The puck is then swung in a circle. What force is producing the centripetal acceleration of the puck? A. Gravity B. Air resistance C. Friction D. Normal force E. Tension in the string

E. Tension in the string

When a car turns a corner on a level road, which force provides the necessary centripetal acceleration?

Friction

A car is turning a corner at a constant speed, following a segment of a circle. What force provides the necessary centripetal acceleration?

Friction - static Imagine you are driving a car on a frictionless road, such as a very icy road. You would not be able to turn a corner. Turning the steering wheel would be of no use. The car would slide straight ahead, in accordance with both Newton's first law and the experience of anyone who has ever driven on ice! So it must be friction that causes the car to turn. The top view of the tire in the figure above shows the force on one of the car's tires as it turns a corner.

Centrifugal Force

If you are a passenger in a car that turns a corner quickly, it is the force of the car door, pushing inward toward the center of the curve, that causes you to turn the corner. • What you feel is your body trying to move ahead in a straight line as outside forces (the door) act to turn you in a circle. A centrifugal force will never appear on a freebody diagram and never be included in Newton's laws.

A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the coin's velocity? n-a,s-c, e-b or w-d

North - A

In the track and field event known as the hammer throw, an athlete spins a heavy mass in a circle at the end of a chain. Once the mass gets moving at a good clip, the athlete lets go of the chain. The mass flies off in a parabolic arc; the winner is the one who gets the maximum distance. For male athletes, the "hammer" is a mass of 7.3 kg at the end of a 1.2-m chain. A world-class thrower can get the hammer up to a speed of 29 m/s. If an athlete swings the mass in a horizontal circle centered on the handle he uses to hold the chain, what is the tension in the chain?

T=5116N

A ball at the end of a string is being swung in a horizontal circle. What force is producing the centripetal acceleration of the ball? A. Gravity B. Air resistance C. Normal force D. Tension in the string

Tension in the string

A passenger on a carnival ride rides in a car that spins in a horizontal circle as shown at right. At the instant shown, which arrow gives the direction of the net force on one of the riders?

The arrow pointing to the left, bc it points towards the center of the path and the motion is moving counterclockwise.

A car is turning a tight corner at a constant speed. A top view of the motion is shown in the figure below. The velocity vector for the car points to the east at the instant shown. What is the direction of the acceleration?

The curve that the car is following is a segment of a circle, so this is an example of uniform circular motion. For uniform circular motion, the acceleration is directed toward the center of the circle, which is to the south.

Period of the Motion

The time interval it takes an object to go around a circle (circumference-2pier) one time. T

For an object in uniform circular motion, what can you say about the directions of the velocity, acceleration, and net force vectors?

The velocity vector is perpendicular to the acceleration vector; the acceleration vector is parallel to the net force vector.

Maximum Walking Speed

The weight is larger than the normal force, the Fnet is in between. Newton's Second Law: Fx=w-n=mv*2/r - in this setting, n=0 so it will be w=mg=mvmax*2/r (after eliminating the mass) - vmax*2=gr The maximum possible walking speed is limited by r, the length of the leg, and g, the free-fall acceleration.

Astronauts on the International Space Station feel weightless because

Their apparent weight is zero

A ball at the end of a string is being swung in a horizontal circle. What is the direction of the net force on the ball? A. Tangent to the circle B. Toward the center of the circle C. There is no net force.

Toward the center of circle

In the video, a car rounding a corner and a car going over the crest of a hill are both presented as examples of

Uniform Circular Motion

Speed:

V=2pier-T OR v=2piefr

Apparent Weight in Circular Motion: Bottom

When the passenger on the roller coaster is at the bottom of the loop: - The net force points upward, so n > w. - Her apparent weight is wapp= n, so her apparent weight is greater than her true weight. Newton's 2nd law: fx=nx+wx=n-w=mv*2/r wapp= n=w+mv*2-r Apparent weight at the bottom is greater than her true weight, w.

The circular blade of a table saw is 25 cm in diameter and spins at 3600 rpm. How much time is required for one revolution? How fast is one of the teeth at the edge of the blade moving? What is the tooth's acceleration?

f=6s*-1 T=1/6=.017s v=47m/s a=18000m/s*2

Apparent Weight in Circular Motion

Your sensation of weight changes on roller coasters as you go over crests and through dips. The force you feel is the contact force that supports you.

Instantaneous Acceleration

directed toward the center of the circle at all points.

A hard drive disk rotates at 7200 rpm. The disk has a diameter of 5.1 in (13 cm). What is the speed of a point 6.0 cm from the center axle? What is the acceleration of this point on the disk?

f=120s T=.0083s*-1 v=45.39m/s a=34337m/s*2

Circular Motion: Frequency

how we specify circular motion # of revolutions per second: f=1-T use inverse seconds s*-1

For uniform circular motion, the acceleration __________.

is directed toward the center of the circle

Instantaneous Velocity

is tangent to the circle at all times

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?

normal force of the road on the car greater than car's weight Because a points upward, by Newton's second law there must be a net force on the car that also points upward. In order for this to be the case, the free-body diagram shows that the magnitude of the normal force must be greater than the weight.

Apparent Weight in Circular Motion: Top

nx+wx=n+w=mv*2/r The w is positive because the x-axis is directed downward. wapp= n=mv*2/r - w If v is sufficiently larger, her apparent weight can exceed the true weight, just like at the bottom of the loop.

A handful of professional skaters have taken a skateboard through an inverted loop in a full pipe. For a typical pipe with a diameter 14 feet, what is the minimum speed the skater must have at the very top of the loop?

r=7ft v=4.56m/s

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?

v=14m/s With no friction acting, it is the horizontal component of the normal force that causes the centripetal acceleration.

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.)

vmax=14m/s Assess: The car's mass canceled out. The maximum speed DOES NOT depend on the mass of the vehicle. The expression for vmax DOES depend on the coefficient of friction and the radius of the turn. The speed at which you can take a turn decrease if us is less or if r is smaller.

A level curve on a country road has a radius of 150 m. What is the maximum speed at which this curve can be safely negotiated on a rainy day when the coefficient of friction between the tires on a car and the road is 0.40?

vmax=24m/s

Velocity and Acceleration in Uniform Motion

• Although the speed of a particle in uniform circular motion is constant, its velocity is not constant because the direction of the motion is always changing. a=v*2/r