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21.E: Applications of Newton's Laws (Exercises)

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    Conceptual Questions

    1. If you wish to reduce the stress (which is related to centripetal force) on high-speed tires, would you use large- or small-diameter tires? Explain.
    2. Define centripetal force. Can any type of force (for example, tension, gravitational force, friction, and so on) be a centripetal force? Can any combination of forces be a centripetal force?
    3. If centripetal force is directed toward the center, why do you feel that you are ‘thrown’ away from the center as a car goes around a curve? Explain.
    4. Race car drivers routinely cut corners, as shown below (Path 2). Explain how this allows the curve to be taken at the greatest speed.

    Two paths are shown inside a race track through a ninety degree curve. Two cars, a red and a blue one, and their paths of travel are shown. The blue car is making a tight turn on path one, which is the inside path along the track. The red car is shown overtaking the first car, while taking a wider turn and crossing in front of the blue car into the inside path and then back out of it.

    1. Many amusement parks have rides that make vertical loops like the one shown below. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if:
      1. The car goes over the top at faster than this speed?
      2. The car goes over the top at slower than this speed?

    A photo of a roller coaster with a vertical loop. The loop has a tighter curvature at the top than at the bottom, making an inverted teardrop shape.

    1. What causes water to be removed from clothes in a spin-dryer?
    2. As a skater forms a circle, what force is responsible for making his turn? Use a free-body diagram in your answer.
    3. Suppose a child is riding on a merry-go-round at a distance about halfway between its center and edge. She has a lunch box resting on wax paper, so that there is very little friction between it and the merry-go-round. Which path shown below will the lunch box take when she lets go? The lunch box leaves a trail in the dust on the merry-go-round. Is that trail straight, curved to the left, or curved to the right? Explain your answer.

    An illustration of the circular base of a merry-go-round with a single horse and child on it. The angular velocity, omega, is clockwise, shown here with an arrow. A point P is shown near the horse, on a circle concentric with the merry-go-round. Three arrows are shown coming out of point P, depicting the three possible path of the lunch box. Path A curves into the circle, to the right from the perspective of the box. Path B is straight, tangent to the circle. Path C curves to the left from the perspective of the box, out of the circle.

    1. Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your car’s speed? What is the direction of the force exerted on you by the car seat?
    2. Suppose a mass is moving in a circular path on a frictionless table as shown below. In Earth’s frame of reference, there is no centrifugal force pulling the mass away from the center of rotation, yet there is a force stretching the string attaching the mass to the nail. Using concepts related to centripetal force and Newton’s third law, explain what force stretches the string, identifying its physical origin.

    An illustration of a mass moving in a circular path on a table. The mass is attached to a string that is pinned at the center of the circle to the table at the other end.

    1. When a toilet is flushed or a sink is drained, the water (and other material) begins to rotate about the drain on the way down. Assuming no initial rotation and a flow initially directly straight toward the drain, explain what causes the rotation and which direction it has in the Northern Hemisphere. (Note that this is a small effect and in most toilets the rotation is caused by directional water jets.) Would the direction of rotation reverse if water were forced up the drain?
    2. A car rounds a curve and encounters a patch of ice with a very low coefficient of kinetic fiction. The car slides off the road. Describe the path of the car as it leaves the road.
    3. In one amusement park ride, riders enter a large vertical barrel and stand against the wall on its horizontal floor. The barrel is spun up and the floor drops away. Riders feel as if they are pinned to the wall by a force something like the gravitational force. This is an inertial force sensed and used by the riders to explain events in the rotating frame of reference of the barrel. Explain in an inertial frame of reference (Earth is nearly one) what pins the riders to the wall, and identify all forces acting on them.
    4. 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/s2. Who do you agree with and why?
    5. 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?

    Problems

    1. (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.
    2. 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.
    3. 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. What is the ideal speed to take a 100.0-m-radius curve banked at a 20.0° angle?
    5. (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?
    6. 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 \(\theta\) (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, \(\theta\) = tan−1 \(\left(\dfrac{v^{2}}{rg}\right)\). (b) Calculate \(\theta\) 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.

    1. 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?
    2. 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. 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?

    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.

    1. 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.

    1. 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 x 10−11 m, and the speed of the electron is 2.18 x 106 m/s. The mass of an electron is 9.11 x 10−31 kg. What is the force on the electron?
    2. 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?
    3. The CERN particle accelerator is circular with a circumference of 7.0 km. (a) What is the acceleration of the protons (m = 1.67 x 10−27 kg) that move around the accelerator at 5% of the speed of light? (The speed of light is v = 3.00 x 108 m/s.) (b) What is the force on the protons?
    4. 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?
    5. 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?
    6. A stunt cyclist rides on the interior of a cylinder 12 m in radius. (They are riding on the side of the cylinder.) The coefficient of static friction between the tires and the wall is 0.68. Find the value of the minimum speed for the cyclist to perform the stunt.
    7. A plumb bob hangs from the roof of a railroad car. The car rounds a circular track of radius 300.0 m at a speed of 90.0 km/h. At what angle relative to the vertical does the plumb bob hang?
    8. A large centrifuge, like the one shown below, 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 10g if the rider is 15.0 m from the center of rotation? (b) The rider’s cage hangs on a pivot at the end of the arm, allowing it to swing outward during rotation as shown in the bottom accompanying figure. At what angle \(\theta\) below the horizontal will the cage hang when the centripetal acceleration is 10g? (Hint: The arm supplies centripetal force and supports the weight of the cage. Draw a free-body diagram of the forces to see what the angle \(\theta\) should be.)

    (a) A photograph of a high g training centrifuge. The astronaut sits in a cage at the end of a long arm that rotates in a horizontal plane. (b) An illustration of a top view of the centrifuge along with an illustration of the forces. The free body diagram shows the weight, w, pointing vertically down and the force F sub arm pointing up and to the left. The forces are then shown rearranged to form a right triangle. F sub arm is the hypotenuse of the triangle pointing up and left, w is the vertical side pointing down, and F sub c is the base pointing to the left. The F sub c arrow is then shown separately with the notation that vector F sub c equals F sub net.

    1. An airplane flying at 200.0 m/s makes a turn that takes 4.0 min. What bank angle is required? What is the percentage increase in the perceived weight of the passengers?

    Contributors and Attributions

    Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).

    This page titled 21.E: Applications of Newton's Laws (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.