第6章ニュートンの法則の応用

(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.

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.

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?

What is the ideal speed to take a 100.0-m-radius curve banked at a $20.0\text{°}$ angle?

(a) What is the radius of a bobsled turn banked at $75.0\text{°}$ 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?

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, $\theta ={\text{tan}}^{\mathrm{-1}}(v^2\text{/}rg).$ (b) Calculate $\theta$ for a 12.0-m/s turn of radius 30.0 m (as in a race).

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 $15.0\text{°}$. (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km/h?

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?

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?

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 $5.28\times {10}^{\mathrm{-11}}\text{m,}$ and the speed of the electron is $2.18\times {10}^6\text{m}\text{/}\text{s}.$ The mass of an electron is $9.11\times {10}^{\mathrm{-31}}\text{kg}$. What is the force on the electron?

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

The CERN particle accelerator is circular with a circumference of 26.7 km. (a) What is the acceleration of the protons $(m=1.67\times {10}^{\mathrm{-27}}\text{kg})$ that move around the accelerator at $5\%$ of the speed of light? (The speed of light is $v=3.00\times {10}^8\text{m/s}\text{.}$) (b) What is the force on the protons?

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 can traverse the curve without slipping?

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?