Circular Motion and Gravitation: Extra Problems

Problem Set A

Problem 1:

Determine the speed (in m/s) of a car which travels around a 47.7-m diameter circle in a time of 18.82 seconds.

Problem 2:

A car moving at a speed of 65.3 mi/hr moves around a circle with a diameter of 39.2 meters. Determine the acceleration (in m/s/s) of the car. (GIVEN: 2.24 mi/hr = 1.00 m/s)

Problem 3:

Determine the acceleration (in m/s/s) of a rider on the Cajun Cliffhanger (a barrel ride at an amusement park) if the rider makes 7.4 revolutions around the 6.53-m diameter circle in 26.1 seconds.

Problem 4:

An eraser is tied to a string and swung in a circle with a radius of 0.887 meters. The eraser makes 104.9 revolutions in a minute. Determine its acceleration (in m/s/s).

Problem 5:

A tire 0.53 m in radius rotates at a constant rate of 226 revolutions per minute. Find the speed (in m/s) of a small stone lodged in the tread of the tire (on its outer edge).

Problem 6:

(Referring to the previous problem.) Determine the acceleration (in m/s/s) of the stone.

Problem 7:

Young David who slew Goliath experimented with slings before tackling the giant. He found that with a sling of length 0.630 m, he could develop a rotation rate of 8.34 rev/s in his weapon. If he increased the length to 0.926 m, he could revolve the sling only 6.28 times per second. What is the centripetal acceleration (in m/s/s) at 8.34 revolutions per second?

Problem 8:

(Referring to the previous problem.) What is the centripetal acceleration (in m/s/s) at 6.28 revolutions per second?

Problem 9:

It has been suggested that rotating cylinders several miles in length and several miles in diameter be placed in space and used as colonies. Inhabitants of the space colonies would live on the inside surface of the cylinder. Inertial effects would resemble gravity's influence and keep them 'plastered to the surface.' Suppose that you are an inhabitant of a space colony which is 10.4 miles in length and 5.2 miles in diameter. How many revolutions per hour must the cylinder have in order for the occupants to experience a centripetal acceleration equal to the acceleration of gravity? HINT: Begin by finding the tangential velocity and convert to revolutions per hour. Given: 1609 m = 1 mile

Problem 10:

A person is flying a 82.2-g model airplane in a horizontal circular path on the end of a string 10.5 m long. The string is horizontal and exerts a force of 3.22 N on the hand of the person holding it. What is the speed (in m/s) of the plane?

Problem 11:

A car of mass 1729 kg rounds a circular turn of radius 20.4 m. The road is flat and the coefficient of friction is 0.764 between the tires and the road. How fast (in m/s) can the car go without skidding?

Problem 12:

Tarzan (m = 78.4 kg) tries to cross a river by swinging from a vine. The vine is 10.0 m long. Tarzan doesn't know that the vine has a breaking strength of 1045 N. What maximum speed (in m/s) can Tarzan have at the bottom of the swing (as he just clears the water) in order to safely cross the river without breaking the vine?

Problem 13:

An airplane is flying in a horizontal circle at a speed of 117 m/s. The 78.4-kg pilot does not want his centripetal accceleration to exceed 6.8g (that is, to be more than 6.8 times the acceleration of gravity). What is the minimum radius (in meters) of the circular path?

Problem 14:

(Referring to the previous problem.) At this radius, what is the net centripetal force (in Newtons) (exerted by the seat belts, friction between the pilot and the seat, and so forth)?

Problem 15:

In a swinging mishap, Tarzan, whose mass is 78.4 kg, finds himself circling a tree in a horizontal circle at the end of a vine 2.95 m long that makes an angle of 5.64 degrees with the vertical. Find the centripetal force (in Newtons) exerted on him by the vine.

Problem 16:

(Referring to the previous problem.) Determine Tarzan's centripetal acceleration (in m/s/s).

Problem 17:

A roller coaster vehicle has a mass of 522 kg when fully loaded with passengers (see diagram at right). If the vehicle has a speed of 24.9 m/s at point A, what is the force (in Newtons) of the track on the vehicle at this point? (Given: R₁ = 11.1 m; R₂ = 16.7 m.)

Problem 18:

(Referring to the previous problem.) What is the maximum speed (in m/s) the vehicle can have at point B in order that it remain on the track?

Problem 19:

A pail of water is rotated in a vertical circle of radius 1.045 meters. What must be the minimum speed (in m/s) of the pail at the top of the circle in order for no water to spill out?

Problem 20:

A 653 kg roller coaster car (includes mass of occupants) are passing through a vertical loop. The speed of the car at the top of the loop is 16.9 m/s. What radius of curvature (in meters) must the loop have at its very top in order for the occupants to experience a normal force which is 1/4-th their weight?

Problem Set B

Problem 1:

Consider two celestial bodies of mass 4.77 x 10¹⁹ kg and 6.51 x 10²² kg separated by a distance of 5.03 x 10¹⁴ m. Determine the force (in Newtons) of gravitational attraction between these two bodies.

Problem 2:

Suppose that planet X had a mass exactly the same as planet Earth. A person weighs 677 N on Earth. What would be the mass (in kg) of the 677-N person on planet X if its radius were 3.03 times greater than that of planet Earth?

Problem 3:

Use the Table of Planetary Data to find the gravitational force (in Newtons) of attraction between the Sun and the Earth. .

Problem 4:

Two students sitting in adjacent seats in a lecture room have weights of 581 N and 649 N. Assume that Newton's law of universal gravitation can be applied to these two students and find the gravitational force (in Newtons) that one student exerts on the other when they are separated by 0.516 m. 

Problem 5:

A 31400-kg spaceship is halfway betwen the Earh and the Moon. Use the Table of Planetary Data to find the net gravitational force (in Newtons) of attraction exerted on the ship by the Earth and the Moon. (Given: Earth-moon distance = 3.84 x 10⁸ m)

Problem 6:

How many Earth radii above the Earth (not from its center) must you be located to experience an acceleration of gravity of 2.378 m/s/s. Express in terms of Earth-radii; that is, express the answer as the number of times greater than 6.38 x 10⁶ m. 

Problem 7:

Use the Table of Planetary Data to find the acceleration of gravity (in m/s/s) at the surface of Jupiter.

Problem 8:

Use the Table of Planetary Data to find the acceleration of gravity (in m/s/s) at the surface of Venus.

Problem 9:

A satellite is in a circular orbit 803 kilometers above the surface of the Moon. Use the Table of Planetary Data for the mass and radius of the Moon. What is the acceleration (in m/s/s) of the satellite? 

Problem 10:

(Referring to the previous problem.) What is the speed (in km/s) of the satellite?

Problem 11:

(Referring to the previous problem.) What is the orbital period (in hours) of the satellite?

Problem 12:

A communication satellite is placed at an altitude of 24036 miles above the surface of the Earth. Find the acceleration (in m/s/s) due to gravity at this altitude. (Given: 1609 m = 1 mile) 

Problem 13:

A satellite of mass 427 kg is in circular orbit about the Earth at a height above the Earth equal to 0.95 times the mean radius of the earth. Find the satellite's orbital speed (in m/s).

Problem 14:

(Referring to the previous problem.) Find the satellite's orbital period (in minutes).

Problem 15:

(Referring to the previous problem.) Find the gravitational force (in Newtons) acting upon the satellite. Express your answer using scientific notation.

Problem 16:

Consider two celestial bodies separated by a distance of 4.82x10⁹ meters. The mass of one body is 4.01x10⁶ kg and the mass of the other body is 6.16x10⁶ kg. Determine the distance (in meters) from the 4.01x10⁶-kg body at which the net gravitational force on an object will be zero. 

Problem 17:

A large planet has several moons. One of the moons has an orbital period of 1.55 days and an orbital radius of 3.13 x 10⁵ km. From this data, determine the mass (in kg) of the planet. 

Problem 18:

Determine the acceleration of gravity (in m/s/s) at an altitude of 2181 miles above the surface of the moon. The moon's radius and mass are listed in Table 7.3 on p. 196. (Given: 1609 m = 1 mile) Enter your answer accurate to the fourth decimal place.

Problem 19:

Determine the orbital period (in minutes) of a 314-kg satellite orbiting Mars at an altitude of 2 Mars-radii. Use the Table of Planetary Data.

Problem 20:

What would be the orbital speed (in mi/hr) of a 331-kg satellite orbitting Earth at an altitude of 1355 miles above the surface of the Earth. (1.0 m/s = 2.24 mi/hr) 

Problem Set C

Problem 1:

A planet is known to have several moons which orbit in circular motion about the planet. One such moon is known to have a period of 2.25 years. Another moon is 3.28 times further from the planet's center. What is its orbital period (in years)?

Problem 2:

A planet has a single moon which orbits at a distance of 7.25x10⁶ meters from the planet's center and makes a revolution in 2.34 years. Determine the mass (in kg) of the planet. 

Problem 3:

A highway curve has a radius of 126 m and is designed for a traffic speed of 31.7 mi/hr (14.1 m/s). If the curve is not banked, determine the minimum coefficient of friction needed between a car and the road to keep a car from skidding. 

Problem 4:

An athlete swings a ball of mass 3.84 kg horizontally on the end of a rope. The ball moves in a circle of radius 0.706 m at an angular speed of 0.546 revolutions/second. What is the tangential velocity (in m/s) of the ball?

Problem 5:

(Referring to the previous problem.) If the maximum tension that the rope can withstand before breaking is 78.9 N, then what is the maximum tangential velocity (in m/s) that the ball can have?

Problem 6:

The Solar Maximum Satellite was placed in a circular orbit of about 150 miles above the Earth. Determine the orbital speed (in m/s) of the satellite. Use data from the Table of Planetary Data. (Given: 1609 m = 1 mile)

Problem 7:

(Referring to the previous problem.) Determine the time (in hours) for one complete revolution.

Problem 8:

A 1268-kg car moves across a bridge made in the shape of a circular arc of radius 30.4 m at a speed of 10.8 m/s. Find the normal force (in Newtons) acting upon the car when it is at the top of its arc.

Problem 9:

(Referring to the previous problem.) At what speed (in m/s) will the normal force reduce to a value of 0 Newtons? (HINT: The normal force becomes zero when the car loses contact with the road.)

Problem 10:

A massive ball is attached to a string and twirled in a horizontal circle with a radius of  1.2-meters. The horizontal component of tension on the string is 36.7 Newtons and the ball takes 0.75 seconds to complete one revolution. Determine the mass (in kg) of the ball.

Problem 11:

A 2.53-gram penny is on a turntable. It starts from rest at a radius of 2.22 meters. The turntable accelerates to a speed of 2.84 m/s, at which time the penny slides off the turntable. What coefficient of static friction exists between the penny and the turntable if the penny doesn't fly off until it reaches this speed? 

Problem 12:

Planet Zwork orbits a massive sun much as the planets of our solar system orbit our Sun. It takes planet Zwork 4.46 years to orbit the sun. Planet Zwork is a distance of 1.22x10¹² meters from the center of its sun. Determine the period (in years) for planet Kep to orbit if the distance from its center to the center of the planet's sun is 6.87x10¹² meters.

Problem 13:

Irada Inavator is standing in an elevator when it begins to accelerate upward. With a 42-kg mass, Irada begins to accelerate upward at 0.99 m/s/s. If Irada were standing upon a Newton force scale as the elevator accelerates upward, then what force (in Newtons) would the scale read?

Problem 14:

A 655-kg roller coaster car starts from the top of a hill and rolls down. It enters a loop for which the radius at the top is 7.57 meters. Determine the minimum speed in meters/second (at the loop's top) at which the 655 kg roller coaster car will complete the loop without falling out of the loop. (HINT: This is the speed at which the roller coaster car wheels are just barely in contact with the track; any slower speed would turn the car into a projectile.)

Problem 15:

A 1.52-gram airplane is attached to a string with a length of 1.05 meters. The plane flies in a circle in which the string is declined at an angle (Theta) of 16.9 degrees below the horizontal. At what speed (in m/s) is the plane flying?