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Chapter 5: Problem 41

A car approaches the top of a hill that is shaped like a vertical circle with a radius of \(55.0 \mathrm{m} .\) What is the fastest speed that the car can go over the hill without losing contact with the ground?

Short Answer

Expert verified
Answer: The maximum speed the car can go over the hill without losing contact with the ground is approximately 26.1 m/s.

Step by step solution

01

Problem Data

We have a hill shaped like a vertical circle with a radius of 55.0 m. Let's denote the speed of the car as v.
02

Centripetal Force and Gravitational Force

The centripetal force acting on a body moving in a circle is \(F_c = \dfrac{mv^2}{r}\), where m is the mass of the car, v is its speed, and r is the radius of the circle (55.0 m in this case). The gravitational force acting on the car is \(F_g = mg\), where g is the acceleration due to gravity (approximately \(9.8 \,\text{m/s}^2\)).
03

Critical Condition for Loss of Contact

The car will lose contact with the ground when the centripetal force is equal to the gravitational force, as they both act on the same object. The normal force acting between the car's tires and the ground decreases to zero when the car loses contact. Thus, we can write the equation for the critical condition as: \(F_c = F_g\) or \(\dfrac{mv^2}{r} = mg\).
04

Mass Cancellation and Calculating Speed

Notice that the mass m appears on both sides of the equation, so we can cancel it out: \(\dfrac{v^2}{r} = g\). Now we just need to solve for v. Rearrange the equation to find v: \(v^2 = rg\) and then take the square root of both sides to get: \(v = \sqrt{rg}\).
05

Plugging in Values and Solving for V

Now we can plug in the known values for r and g: \(v = \sqrt{(55.0 \, \text{m})(9.8 \, \text{m/s}^2)}\). Calculate the value for v to get the fastest speed the car can go over the hill without losing contact with the ground: \(v \approx 26.1 \, \text{m/s}\). Thus, the fastest speed that the car can go over the hill without losing contact with the ground is \(\boxed{26.1\,\text{m/s}}\).

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Most popular questions from this chapter

A person rides a Ferris wheel that turns with constant angular velocity. Her weight is \(520.0 \mathrm{N}\). At the top of the ride her apparent weight is \(1.5 \mathrm{N}\) different from her true weight. (a) Is her apparent weight at the top \(521.5 \mathrm{N}\) or \(518.5 \mathrm{N} ?\) Why? (b) What is her apparent weight at the bottom of the ride? (c) If the angular speed of the Ferris wheel is \(0.025 \mathrm{rad} / \mathrm{s},\) what is its radius?
A child swings a rock of mass \(m\) in a horizontal circle using a rope of length \(L\). The rock moves at constant speed \(v\). (a) Ignoring gravity, find the tension in the rope. (b) Now include gravity (the weight of the rock is no longer negligible, although the weight of the rope still is negligible). What is the tension in the rope? Express the tension in terms of \(m, g, v, L,\) and the angle \(\theta\) that the rope makes with the horizontal.
The Milky Way galaxy rotates about its center with a period of about 200 million yr. The Sun is \(2 \times 10^{20} \mathrm{m}\) from the center of the galaxy. (a) What is the Sun's radial acceleration? (b) What is the net gravitational force on the Sun due to the other stars in the Milky Way?
Objects that are at rest relative to the Earth's surface are in circular motion due to Earth's rotation. (a) What is the radial acceleration of an object at the equator? (b) Is the object's apparent weight greater or less than its weight? Explain. (c) By what percentage does the apparent weight differ from the weight at the equator? (d) Is there any place on Earth where a bathroom scale reading is equal to your true weight? Explain.
Centrifuges are commonly used in biological laboratories for the isolation and maintenance of cell preparations. For cell separation, the centrifugation conditions are typically \(1.0 \times 10^{3}\) rpm using an 8.0 -cm-radius rotor. (a) What is the radial acceleration of material in the centrifuge under these conditions? Express your answer as a multiple of \(g .\) (b) At $1.0 \times 10^{3}$ rpm (and with a 8.0-cm rotor), what is the net force on a red blood cell whose mass is \(9.0 \times 10^{-14} \mathrm{kg} ?\) (c) What is the net force on a virus particle of mass \(5.0 \times 10^{-21} \mathrm{kg}\) under the same conditions? (d) To pellet out virus particles and even to separate large molecules such as proteins, superhigh-speed centrifuges called ultracentrifuges are used in which the rotor spins in a vacuum to reduce heating due to friction. What is the radial acceleration inside an ultracentrifuge at 75000 rpm with an 8.0 -cm rotor? Express your answer as a multiple of \(g\).
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