Gravity

Dawson High School AP Physics 1Uniform Circular Motion and GravityAcknowledgements Mark Lesmeister/Pearland ISD

Selected questions from Cutnell and Johnson,Physics 9e Instructors Resource Site, 2014 Wiley and Sons.

2Uniform Circular MotionPart 1Centripetal accelerationPhysical ResultEveryday ConfirmationAn object moving in a circular path is accelerating, even if it moves with constant speed.The acceleration is directly proportional to the square of the speed.The acceleration is inversely proportional to the radius of the circle.A force is required to keep something moving in a circle.On a slick road, we need to slow down before turning.

We feel more centrifugal force in a tight turn at the same speed. Centripetal Acceleration

v= speed of the object moving in a circle.r= radius of the circle.

The direction of the acceleration is toward the center of the circle.

vaSpeed and Period

vaCentripetal ForceThis force is not a new force. It just tells us the amount of force that must be provided, by a tension, gravity, etc., in order for an object to move in a circle.

vaCircular Motion Example 1A pail of water is swing in a circular path of radius r and a speed at the top of vt. Find the force exerted by the pail on the water.Find the minimum speed with which the bucket can be swung and still have the water remain in the pail at the top.

FPWDemonstrate.8

Circular Motion Example 2A stock-car is driven on a curved, banked track at a speed v. If the radius of curvature is r and the bank angle is , find an expression for the angle that will allow the car to go around the turn even if the road is frictionless.

Circular Motion Example 2A stock-car is driven on a curved, banked track at a speed v. If the radius of curvature is r and the bank angle is , find an expression for the angle that will allow the car to go around the turn even if the road is frictionless.Hint- Dont use the parallel and perpendicular to the plane directions, use horizontal and vertical.

Circular Motion Example 3A car is moving on a flat circular track, with a radius of 100 m and a coefficient of static friction s = 0.4. How fast can the car go around the track?5.1.1. An airplane flying at 115 m/s due east makes a gradual turn while maintaining its speed and follows a circular path to fly south. The turn takes 15 seconds to complete. What is the radius of the circular path?

a) 410 m

b) 830 m

c) 1100 m

d) 1600 m

e) 2200 m5.2.1. A ball is whirled on the end of a string in a horizontal circle of radius R at constant speed v. By which one of the following means can the centripetal acceleration of the ball be increased by a factor of two?

a) Keep the radius fixed and increase the period by a factor of two.

b) Keep the radius fixed and decrease the period by a factor of two.

c) Keep the speed fixed and increase the radius by a factor of two.

d) Keep the speed fixed and decrease the radius by a factor of two.

e) Keep the radius fixed and increase the speed by a factor of two.5.2.2. A steel ball is whirled on the end of a chain in a horizontal circle of radius R with a constant period T. If the radius of the circle is then reduced to 0.75R, while the period remains T, what happens to the centripetal acceleration of the ball?

a) The centripetal acceleration increases to 1.33 times its initial value.

b) The centripetal acceleration increases to 1.78 times its initial value.

c) The centripetal acceleration decreases to 0.75 times its initial value.

d) The centripetal acceleration decreases to 0.56 times its initial value.

e) The centripetal acceleration does not change.5.2.3. While we are in this classroom, the Earth is orbiting the Sun in an orbit that is nearly circular with an average radius of 1.50 1011 m. Assuming that the Earth is in uniform circular motion, what is the centripetal acceleration of the Earth in its orbit around the Sun?

a) 5.9 103 m/s2

b) 1.9 105 m/s2

c) 3.2 107 m/s2

d) 7.0 102 m/s2

e) 9.8 m/s25.2.4. A truck is traveling with a constant speed of 15 m/s. When the truck follows a curve in the road, its centripetal acceleration is 4.0 m/s2. What is the radius of the curve?

a) 3.8 m

b) 14 m

c) 56 m

d) 120 m

e) 210 m5.2.5. Consider the following situations:(i) A minivan is following a hairpin turn on a mountain road at a constant speed of twenty miles per hour.(ii) A parachutist is descending at a constant speed 10 m/s.(iii) A heavy crate has been given a quick shove and is now sliding across the floor.(iv) Jenny is swinging back and forth on a swing at the park.(v) A football that was kicked is flying through the goal posts.(vi) A plucked guitar string vibrates at a constant frequency.In which one of these situations does the object or person experience zero acceleration?

a) i only

b) ii only

c) iii and iv only

d) iv, v, and vi only

e) all of the situations5.3.1. A boy is whirling a stone at the end of a string around his head. The string makes one complete revolution every second, and the tension in the string is FT. The boy increases the speed of the stone, keeping the radius of the circle unchanged, so that the string makes two complete revolutions per second. What happens to the tension in the sting?

a) The tension increases to four times its original value.

b) The tension increases to twice its original value.

c) The tension is unchanged.

d) The tension is reduced to one half of its original value.

e) The tension is reduced to one fourth of its original value.5.3.2. An aluminum rod is designed to break when it is under a tension of 600 N. One end of the rod is connected to a motor and a 12-kg spherical object is attached to the other end. When the motor is turned on, the object moves in a horizontal circle with a radius of 6.0 m. If the speed of the motor is continuously increased, at what speed will the rod break? Ignore the mass of the rod for this calculation.

a) 11 m/s

b) 17 m/s

c) 34 m/s

d) 88 m/s

e) 3.0 102 m/s5.3.3. A ball is attached to a string and whirled in a horizontal circle. The ball is moving in uniform circular motion when the string separates from the ball (the knot wasnt very tight). Which one of the following statements best describes the subsequent motion of the ball?

a) The ball immediately flies in the direction radially outward from the center of the circular path the ball had been following.

b) The ball continues to follow the circular path for a short time, but then it gradually falls away.

c) The ball gradually curves away from the circular path it had been following.

d) The ball immediately follows a linear path away from, but not tangent to the circular path it had been following.

e) The ball immediately follows a line that is tangent to the circular path the ball had been following5.3.4. A rancher puts a hay bail into the back of her SUV. Later, she drives around an unbanked curve with a radius of 48 m at a speed of 16 m/s. What is the minimum coefficient of static friction for the hay bail on the floor of the SUV so that the hay bail does not slide while on the curve?

a) This cannot be determined without knowing the mass of the hay bail.

b) 0.17

c) 0.33

d) 0.42

e) 0.545.3.5. Imagine you are swinging a bucket by the handle around in a circle that is nearly level with the ground (a horizontal circle). What is the force, the physical force, holding the bucket in a circular path?

a) the centripetal force

b) the centrifugal force

c) your hand on the handle

d) gravitational force

e) None of the above are correct.5.3.6. Imagine you are swinging a bucket by the handle around in a circle that is nearly level with the ground (a horizontal circle). Now imagine there's a ball in the bucket. What keeps the ball moving in a circular path?

a) contact force of the bucket on the ball

b) contact force of the ball on the bucket

c) gravitational force on the ball

d) the centripetal force

e) the centrifugal force5.3.7. The moon, which is approximately 4 109 m from Earth, has a mass of 7.4 1022 kg and a period of 27.3 days. What must is the magnitude of the gravitational force between the Earth and the moon?

a) 1.8 1018 N

b) 2.1 1022 N

c) 1.7 1013 N

d) 5.0 1022 N

e) 4.2 1020 N5.3.8. The Rapid Rotor amusement ride is spinning fast enough that the floor beneath the rider drops away and the rider remains in place. If the Rotor speeds up until it is going twice as fast as it was previously, what is the effect on the frictional force on the rider?

a) The frictional force is reduced to one-fourth of its previous value.

b) The frictional force is the same as its previous value.

c) The frictional force is reduced to one-half of its previous value.

d) The frictional force is increased to twice its previous value.

e) The frictional force is increased to four times its previous value.

5.4.1. Complete the following statement: The maximum speed at which a car can safely negotiate an unbanked curve depends on all of the following factors except

a) the coefficient of kinetic friction between the road and the tires.

b) the coefficient of static friction between the road and the tires.

c) the acceleration due to gravity.

d) the diameter of the curve.

e) the ratio of the static frictional force between the road and the tires and the normal force exerted on the car.5.4.2. A 1000-kg car travels along a straight portion of highway at a constant velocity of 10 m/s, due east. The car then encounters an unbanked curve of radius 50 m. The car follows the curve traveling at a constant speed of 10 m/s while the direction of the car changes from east to south. What is the magnitude of the acceleration of the car as it travels the unbanked curve?

a) zero m/s2

b) 2 m/s2

c) 5 m/s2

d) 10 m/s2

e) 20 m/s25.4.3. A 1000-kg car travels along a straight portion of highway at a constant velocity of 10 m/s, due east. The car then encounters an unbanked curve of radius 50 m. The car follows the curve traveling at a constant speed of 10 m/s while the direction of the car changes from east to south. What is the magnitude of the frictional force between the tires and the road as the car negotiates the unbanked curve?

a) 500 N

b) 1000 N

c) 2000 N

d) 5000 N

e) 10 000 N5.4.4. You are riding in the forward passenger seat of a car as it travels along a straight portion of highway. The car continues traveling at a constant speed as it follows a sharp, unbanked curve to the left. You feel the door pushing on the right side of your body. Which of the following forces in the horizontal direction are acting on you?

a) a static frictional force between you and the seat

b) a normal force of the door

c) a force pushing you toward the door

d) answers a and b

e) answers a and c5.5.3. A space probe is orbiting a planet on a circular orbit of radius R and a speed v. The acceleration of the probe is a. Suppose rockets on the probe are fired causing the probe to move to another circular orbit of radius 0.5R and speed 2v. What is the magnitude of the probes acceleration in the new orbit?

a) a/2

b) a

c) 2a

d) 4a

e) 8a5.6.1. A space station is designed in the shape of a large, hollow donut that is uniformly rotating. The outer radius of the station is 460 m. With what period must the station rotate so that a person sitting on the outer wall experiences artificial gravity, i.e. an acceleration of 9.8 m/s2?

a) 43 s

b) 76 s

c) 88 s

d) 110 s

e) 230 s5.7.1. At a circus, a clown on a motorcycle with a mass M travels along a horizontal track and enters a vertical circle of radius r. Which one of the following expressions determines the minimum speed that the motorcycle must have at the top of the track to remain in contact with the track?

a)

b)

c) v = gR

d) v = 2gR

e) v = MgR

5.7.2. A ball on the end of a rope is moving in a vertical circle near the surface of the earth. Point A is at the top of the circle; C is at the bottom. Points B and D are exactly halfway between A and C. Which one of the following statements concerning the tension in the rope is true?

a) The tension is the same at points A and C.

b) The tension is smallest at point C.

c) The tension is smallest at both points B and D.

d) The tension is smallest at point A.

e) The tension is the same at all four points.5.7.3. An aluminum rod is designed to break when it is under a tension of 650 N. One end of the rod is connected to a motor and a 12-kg spherical object is attached to the other end. When the motor is turned on, the object moves in a vertical circle with a radius of 6.0 m. If the speed of the motor is continuously increased, what is the maximum speed the object can have at the bottom of the circle without breaking the rod? Ignore the mass of the rod for this calculation.

a) 4.0 m/s

b) 11 m/s

c) 16 m/s

d) 128 m/s

e) 266 m/s5.7.4. A girl is swinging on a swing in the park. As she wings back and forth, she follows a path that is part of a vertical circle. Her speed is maximum at the lowest point on the circle and temporarily zero m/s at the two highest points of the motion as her direction changes. Which of the following forces act on the girl when she is at the lowest point on the circle?

a) the force of gravity, which is directed downward

b) the force which is directed radially outward from the center of the circle

c) the tension in the chains of the swing, which is directed upward

d) answers b and c only

e) answers a and c only6.5.13. Which of the following parameters determine how fast you need to swing a water bucket vertically so that water in the bucket will not fall out?

a) radius of swing

b) mass of bucket

c) mass of water

d) a and b

e) a and cNewtons Law of GravityPart 2Newtons Law of Universal Gravitation

38Practice ProblemsFind the gravitational force exerted on the moon (mass = 7.36 x 1022 kg) by the Earth (mass = 5.98 x 1024 kg) when the distance between them is 3.84 x 108 m.Given that Earths average distance from the sun (mass = 1.99 x 1030 kg) is 1.50 x 1011 m, find the net gravitational force exerted on the moon by the sun and the Earth during a lunar eclipse (when the Earth is between the sun and the moon.)Gravitational field strengthWe can think of gravity in two ways.One object exerts a force on another object at a distance.An object creates a field that exerts a force on another object.

Gravitational Field Strength (g)The gravitational field strength is the force of gravity per kilogram of mass.Here on the surface of Earth, the gravitational field strength is 9.8 N/kg g = 10 N/kgg = 2.5 N/kgGravitational Field Strength of a Point Mass (or Spherically Symmetric MassThe gravity field around a point mass, or a spherically symmetric mass, depends only on the mass and the distance away from the center.

Finding the speed of an Earth SatelliteCentripetal acceleration:

Centripetal force:

The force is the force of Gravity

The mass of the satellite divides out, so it doesnt matter.

This assumes circular orbits.43Finding the speed of an Earth Satellite (Continued)Solving for v:

This analysis assumes a circular orbit, but it is generalizable to any orbit. 445.5.1. A satellite is in a circular orbit around the Earth. If it is at an altitude equal to twice the radius of the Earth, 2RE, how does its speed v relate to the Earth's radius RE, and the magnitude g of the acceleration due to gravity on the Earth's surface?

a)

b)

c)

d)

e)

5.5.2. It is the year 2094; and people are designing a new space station that will be placed in a circular orbit around the Sun. The orbital period of the station will be 6.0 years. Determine the ratio of the stations orbital radius about the Sun to that of the Earths orbital radius about the Sun. Assume that the Earths obit about the Sun is circular.

a) 2.4

b) 3.3

c) 4.0

d) 5.2

e) 6.0Gravity and Circular Orbits PracticeA satellite is in a orbit around an unknown planet. The satellite has a speed of 1.7 x 104 m/s and the radius of the orbit is 5.25 x 106 m. A second satellite also has a circular orbit around the same planet, with a radius of 8.60 X 106 m. What is the orbital speed of the second satellite?

The Earth travels around the Sun once a year in an approximately circular orbit whose radius is 1.50 x 1011 m. Determine the mass of the Sun. (Hint: First determine the orbital speed of the Earth.) From Cutnell and Johnson, Physics, John Wiley and Sons