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College Physics by Openstax Chapter 6 Problem 30

If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). (a) Calculate the ideal speed to take a 100 m radius curve banked at 15.0º.

The Ideal Speed and the Minimum Coefficient of Friction in Icy Mountain Roads Problem: If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). (a) Calculate the ideal speed to take a 100 m radius curve banked at 15.0º. (b) What is the minimum coefficient of friction needed for…

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College Physics by Openstax Chapter 6 Problem 29

A large centrifuge, like the one shown in Figure 6.34(a), is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries.

The centripetal acceleration of a large centrifuge as experienced in rocket launches and atmospheric reentries of astronauts Problem: A large centrifuge, like the one shown in Figure 6.34(a), is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries. (a) At what angular velocity is the centripetal acceleration  if the rider…

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College Physics by Openstax Chapter 6 Problem 28

Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in Figure 6.33. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved

Riding a Bicycle in an Ideally Banked Curve Problem: Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in Figure 6.33. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the…

[engineeringmath]

College Physics by Openstax Chapter 6 Problem 27

(a) What is the radius of a bobsled turn banked at 75.0° and taken at 30.0 m/s, assuming it is ideally banked? (b) Calculate the centripetal acceleration. (c) Does this acceleration seem large to you?

The radius and centripetal acceleration of a bobsled turn on an ideally banked curve Problem: (a) What is the radius of a bobsled turn banked at 75.0° and taken at 30.0 m/s, assuming it is ideally banked? (b) Calculate the centripetal acceleration. (c) Does this acceleration seem large to you? Solution: Part A For ideally banked curved, the ideal banking angle is given by the formula . We…

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College Physics by Openstax Chapter 6 Problem 26

What is the ideal speed to take a 100 m radius curve banked at a 20.0° angle?

The Ideal Speed on a Banked Curve Problem: What is the ideal speed to take a 100 m radius curve banked at a 20.0° angle? Solution: The formula for the ideal speed on a banked curve can be derived from the formula of the ideal angle. That is, starting from , we can solve for . v = \sqrt{rg \tan \theta} For this problem, we are given the following values: radius of curvature, acceleration…

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College Physics by Openstax Chapter 6 Problem 25

What is the ideal banking angle for a gentle turn of 1.20 km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit?

The ideal banking angle of a curve on a highway Problem: What is the ideal banking angle for a gentle turn of 1.20 km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit? Solution: The ideal banking angle (meaning there is no involved friction) of a car on a curve is given by the formula: \theta = \tan^{-1} \left( \frac{v^2}{rg} \right) We…

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College Physics by Openstax Chapter 6 Problem 24

Calculate the centripetal force on the end of a 100 m (radius) wind turbine blade that is rotating at 0.5 rev/s. Assume the mass is 4 kg.

Centripetal Force of a Rotating Wind Turbine Blade Problem: Calculate the centripetal force on the end of a 100 m (radius) wind turbine blade that is rotating at 0.5 rev/s. Assume the mass is 4 kg. Solution: We are given the following values: radius, angular velocity, mass, Centripetal force  is any force causing uniform circular motion. It is a “center-seeking” force that always points…

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College Physics by Openstax Chapter 6 Problem 23

A 22.0 kg child is riding a playground merry-go-round that is rotating at 40.0 rev/min. What centripetal force must she exert to stay on if she is 1.25 m from its center?

The centripetal force of a child riding a merry-go-round Problem: (a) A 22.0 kg child is riding a playground merry-go-round that is rotating at 40.0 rev/min. What centripetal force must she exert to stay on if she is 1.25 m from its center? (b) What centripetal force does she need to stay on an amusement park merry-go-round that rotates at 3.00 rev/min if she is 8.00 m from its center? (c)…

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Problem 6-21: Centripetal acceleration of an amusement park ride shaped like a Viking ship

Riders in an amusement park ride shaped like a Viking ship hung from a large pivot are rotated back and forth like a rigid pendulum. Sometime near the middle of the ride, the ship is momentarily motionless at the top of its circular arc.

Riders in an amusement park ride shaped like a Viking ship hung from a large pivot are rotated back and forth like a rigid pendulum. Sometime near the middle of the ride, the ship is momentarily motionless at the top of its circular arc. The ship then swings down under the influence of gravity. The speed at the bottom of the arc is 23.4 m/s. (a) What is the centripetal acceleration at the bottom…

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Problem 6-20: The centripetal acceleration of the commercial jet's tires, and the force of a determined bacterium in it

At takeoff, a commercial jet has a 60.0 m/s speed. Its tires have a diameter of 0.850 m. (a) At how many rev/min are the tires rotating? (b) What is the centripetal acceleration at the edge of the tire? (c) With what force must a determined 1.00×10−15

At takeoff, a commercial jet has a 60.0 m/s speed. Its tires have a diameter of 0.850 m. (a) At how many rev/min are the tires rotating? (b) What is the centripetal acceleration at the edge of the tire? (c) With what force must a determined 1.00×10−15 kg bacterium cling to the rim? (d) Take the ratio of this force to the bacterium’s weight. Solution: We are given the following quantities:…

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Problem 6-19: The angular velocity of an "artificial gravity"

A rotating space station is said to create “artificial gravity”—a loosely-defined term used for an acceleration that would be crudely similar to gravity. The outer wall of the rotating space station would become a floor for the astronauts, and centripetal

A rotating space station is said to create “artificial gravity”—a loosely-defined term used for an acceleration that would be crudely similar to gravity. The outer wall of the rotating space station would become a floor for the astronauts, and centripetal acceleration supplied by the floor would allow astronauts to exercise and maintain muscle and bone strength more naturally than in non-rotating…

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Problem 6-18: The linear speed of an ultracentrifuge and Earth in its orbit

Verify that the linear speed of an ultracentrifuge is about 0.50 km/s, and Earth in its orbit is about 30 km/s by calculating: (a) The linear speed of a point on an ultracentrifuge 0.100 m from its center, rotating at 50,000 rev/min.

Verify that the linear speed of an ultracentrifuge is about 0.50 km/s, and Earth in its orbit is about 30 km/s by calculating: (a) The linear speed of a point on an ultracentrifuge 0.100 m from its center, rotating at 50,000 rev/min. (b) The linear speed of Earth in its orbit about the Sun (use data from the text on the radius of Earth’s orbit and approximate it as being…

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Problem 6-17: The acceleration due to gravity at the position of a satellite located above the Earth

What percentage of the acceleration at Earth’s surface is the acceleration due to gravity at the position of a satellite located 300 km above Earth?

What percentage of the acceleration at Earth’s surface is the acceleration due to gravity at the position of a satellite located 300 km above Earth? Solution: The acceleration due to gravity of a body and the Earth is given by the formula g= G \frac{M}{r^2} where is the gravitational constant, is the mass of the Earth, and is the distance of the object to the center of the Earth. We know that…

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Problem 6-16: Calculating the centripetal acceleration of an ice skater's nose

Olympic ice skaters are able to spin at about 5.00 rev/s. (a) What is their angular velocity in radians per second? (b) What is the centripetal acceleration of the skater’s nose if it is 0.120 m from the axis of rotation? (c) An exceptional skater named D

Olympic ice skaters are able to spin at about 5.00 rev/s. (a) What is their angular velocity in radians per second? (b) What is the centripetal acceleration of the skater’s nose if it is 0.120 m from the axis of rotation? (c) An exceptional skater named Dick Button was able to spin much faster in the 1950s than anyone since—at about 9.00 rev/s. What was the centripetal acceleration of the tip…

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Curing the Compulsive Gambler: Challenging Probability Problem

Mr. Brown always bets a dollar on the number 13 at roulette against the advice of Kind Friend. To help cure Mr. Brown of playing roulette, Kind Friend always bets Brown $20 at even money that Brown will be behind at the end of 36 plays.

Mr. Brown always bets a dollar on the number 13 at roulette against the advice of Kind Friend. To help cure Mr. Brown of playing roulette, Kind Friend always bets Brown $20 at even money that Brown will be behind at the end of 36 plays. How is the cure working? (Most American roulette wheels have 38 equally likely numbers. If the player’s number comes up, he is paid 35 times his stake and gets…

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Chuck-a-Luck: Challenging Probability Problem

Chuck-a-Luck is a gambling game often played at carnivals and gambling houses. A player may bet on anyone of the numbers 1, 2, 3, 4, 5, 6. Three dice are rolled. If the player's number appears on one, two, or three of the dice, he receives respectively on

Chuck-a-Luck is a gambling game often played at carnivals and gambling houses. A player may bet on anyone of the numbers 1, 2, 3, 4, 5, 6. Three dice are rolled. If the player’s number appears on one, two, or three of the dice, he receives respectively one, two, or three times his original stake plus his own money back; otherwise, he loses his stake. What is the player’s expected loss per unit…

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Coin in Square: Challenging Probability Problem

In a common carnival game, a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his pe

In a common carnival game, a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise, he loses his penny. If the penny lands on the table, what is his chance to win? Solution: When we toss the coin onto the…