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Two friends are having a conversation. Anna says a satellite in orbit is in free fall because the satellite keeps falling toward Earth. Tom says a satellite in orbit is not in free fall because the acceleration due to gravity is not 9.80 m/s 2 . Who do you agree with and why?

Anna is correct. The satellite is freely falling toward Earth due to gravity, even though gravity is weaker at the altitude of the satellite, and g is not 9.80 m/s 2 . Free fall does not depend on the value of g ; that is, you could experience free fall on Mars if you jumped off Olympus Mons (the tallest volcano in the solar system).

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A nonrotating frame of reference placed at the center of the Sun is very nearly an inertial one. Why is it not exactly an inertial frame?

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Problems

(a) A 22.0-kg child is riding a playground merry-go-round that is rotating at 40.0 rev/min. What centripetal force is exerted if he is 1.25 m from its center? (b) What centripetal force is exerted if the merry-go-round rotates at 3.00 rev/min and he is 8.00 m from its center? (c) Compare each force with his weight.

a. 483 N; b. 17.4 N; c. 2.24, 0.0807

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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.

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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?

4.14 °

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What is the ideal speed to take a 100.0-m-radius curve banked at a 20.0 ° angle?

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(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?

a. 24.6 m; b. 36.6 m/s 2 ; c. 3.73 times g

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Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen below. 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 centripetal force) and the vertical normal force (which must equal the system’s weight). (a) Show that θ (as defined as shown) is related to the speed v and radius of curvature r of the turn in the same way as for an ideally banked roadway—that is, θ = tan −1 ( v 2 / r g ) . (b) Calculate θ for a 12.0-m/s turn of radius 30.0 m (as in a race).

The figure is an illustration of a man riding a bicycle, viewed from the front. The rider and bike are tilted to the right at an angle theta to the vertical. Three force vectors are shown as solid line arrows. One is from the bottom of the front wheel to the right showing the centripetal force F sub c. A second is from the same point vertically upward showing the force N. The third is from the chest of the rider vertically downward showing his weight, w. An additional broken line arrow from the bottom of the wheel to the chest point, at an angle theta to the right of vertical, is also shown and labeled with force F exerting on it.  The vectors F sub c, w and F form a right triangle whose hypotenuse is F. A free-body diagram is also given above the figure showing vectors w and F. The vector relations F equals the sum of N and F sub c, and N equals w are also given alongside the figure.
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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 problem on icy mountain roads). (a) Calculate the ideal speed to take a 100.0 m radius curve banked at 15.0 ° . (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km/h?

a. 16.2 m/s; b. 0.234

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Modern roller coasters have vertical loops like the one shown here. The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into their seats. (a) What is the speed of the roller coaster at the top of the loop if the radius of curvature there is 15.0 m and the downward acceleration of the car is 1.50 g ? (b) How high above the top of the loop must the roller coaster start from rest, assuming negligible friction? (c) If it actually starts 5.00 m higher than your answer to (b), how much energy did it lose to friction? Its mass is 1.50 × 10 3 kg .

An illustration of a loop of a roller. The radius of curvature is smaller at the top than on the sides and bottom. The radius of the loop at the tom is shown and labeled as r sub minimum. The radius at the lowest part of the loop is labeled as r sub maximum.  The track is on the inside surface of the loop. The motion is indicated by arrows, starting at ground level to the right of the loop, going up inside the loop on the left, then down the inside right of the loop, and out again at ground level on the left. Four location on the track, A, B, C, and D and B, are labeled. Point A is at ground level, to the right of the loop, where the track is straight and horizontal. Point B is part way up the left side of the loop. Point C is part way up the right side of the loop, at the same level as point B. Point D is at ground level, to the left of the loop, where the track is straight and horizontal.
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A child of mass 40.0 kg is in a roller coaster car that travels in a loop of radius 7.00 m. At point A the speed of the car is 10.0 m/s, and at point B, the speed is 10.5 m/s. Assume the child is not holding on and does not wear a seat belt. (a) What is the force of the car seat on the child at point A? (b) What is the force of the car seat on the child at point B? (c) What minimum speed is required to keep the child in his seat at point A?

An illustration of a loop of a roller coaster with a child seated in a car approaching the loop. The track is on the inside surface of the loop. Two location on the loop, A and B, are labeled. Point A is at the top of the loop. Point B is down and to the left of A. The angle between the radii to points A and B is thirty degrees.

a. 179 N; b. 290 N; c. 8.3 m/s

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In the simple Bohr model of the ground state of the hydrogen atom, the electron travels in a circular orbit around a fixed proton. The radius of the orbit is 5.28 × 10 −11 m, and the speed of the electron is 2.18 × 10 6 m / s . The mass of an electron is 9.11 × 10 −31 kg . What is the force on the electron?

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Railroad tracks follow a circular curve of radius 500.0 m and are banked at an angle of 5.0 ° . For trains of what speed are these tracks designed?

20.7 m/s

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The CERN particle accelerator is circular with a circumference of 7.0 km. (a) What is the acceleration of the protons ( m = 1.67 × 10 −27 kg ) that move around the accelerator at 5 % of the speed of light? (The speed of light is v = 3.00 × 10 8 m/s . ) (b) What is the force on the protons?

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A car rounds an unbanked curve of radius 65 m. If the coefficient of static friction between the road and car is 0.70, what is the maximum speed at which the car traverse the curve without slipping?

21 m/s

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A banked highway is designed for traffic moving at 90.0 km/h. The radius of the curve is 310 m. What is the angle of banking of the highway?

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Practice Key Terms 6

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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