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Chapter 6: Q6.3-26 PE (page 222)
What is the ideal speed to take a \(100{\rm{ m}}\) radius curve banked at \(20.0^\circ \) angle?
The ideal speed is \(18.9{\rm{ m}}/{\rm{s}}\).
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A mother pushes her child on a swing so that his speed is at the lowest point of this path. The swing is suspended 2.00 mabove the child鈥檚 center of mass.
(a) What is the magnitude of the centripetal acceleration of the child at the low point?
(b) What is the magnitude of the force the child exerts on the seat if his mass is 18.0 kg?
(c) What is unreasonable about these results?
(d) Which premises are unreasonable or inconsistent?
Space debris left from old satellites and their launchers is becoming a hazard to other satellites
(a) Calculate the speed of a satellite in an orbit900 km above Earth鈥檚 surface.
(b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite鈥檚 orbit at an angle ofrelative to Earth. What is the velocity of the rivet relative to the satellite just before striking it?
(c) Given the rivet is 3.00 mmin size, how long will its collision with the satellite last?
(d) If its mass is0.500 g, what is the average force it exerts on the satellite? (e) How much energy in joules is generated by the collision? (The satellite鈥檚 velocity does not change appreciably, because its mass is much greater than the rivet鈥檚.)
A number of amusement parks have rides that make vertical loops like the one shown in Figure. 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: (a) The car goes over the top at faster than this speed? (b)The car goes over the top at slower than this speed?
Newton鈥檚 laws of motion and gravity were among the first to convincingly demonstrate the underlying simplicity and unity in nature. Many other examples have since been discovered, and we now expect to find such underlying order in complex situations. Is there proof that such order will always be found in new explorations?
Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in Figure. 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鈥攆riction parallel to the road (this must supply the centripetal force), and the vertical normal force (which must equal the system鈥檚 weight).
(a) Show that\(\theta \)(as defined in the figure) is related to the speed v and radius of curvature r of the turn in the same way as for an ideally banked roadway鈥攖hat is,\(\theta = {\tan ^{ - 1}}\,{v^2}/rg\)
(b) Calculate \(\theta \) for a \(12.0{\rm{ m}}/{\rm{s}}\) turn of radius \(30.0{\rm{ m}}\) (as in a race).
Figure 6.36 A bicyclist negotiating a turn on level ground must lean at the correct angle鈥攖he ability to do this becomes instinctive. The force of the ground on the wheel needs to be on a line through the center of gravity. The net external force on the system is the centripetal force. The vertical component of the force on the wheel cancels the weight of the system while its horizontal component must supply the centripetal force. This process produces a relationship among the angle \(\theta \), the speed \(v\), and the radius of curvature \(r\) of the turn similar to that for the ideal banking of roadways.
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