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Chapter 5: Problem 47

A motorcycle is traveling up one side of a hill and down the other side. The crest of the hill is a circular arc with a radius of \(45.0 \mathrm{m}\). Determine the maximum speed that the cycle can have while moving over the crest without losing contact with the road.

Short Answer

Expert verified
The maximum speed is approximately 21.0 m/s.

Step by step solution

01

Understand the Problem

The problem involves a motorcycle traveling over a hill, which forms a circular arc. You need to find the maximum speed at which the motorcycle can travel without losing contact with the road at the crest (top) of the hill. This means, at the maximum speed, the normal force should be zero.
02

Apply Circular Motion Concepts

The condition of the motorcycle losing contact implies the normal force is zero. At the crest, the only forces acting on the motorcycle are its weight and the centripetal force needed to maintain circular motion. Hence, at the maximum speed, the weight of the motorcycle provides the necessary centripetal force.
03

Use the Centripetal Force Equation

The centripetal force required to keep the motorcycle moving in a circle is given by the equation \( F_c = \frac{mv^2}{r} \), where \( m \) is the mass of the motorcycle, \( v \) is the velocity, and \( r \) is the radius of the circle.
04

Equate Weight to Centripetal Force

At the crest, the weight of the motorcycle (\( mg \)) provides the necessary centripetal force for circular motion. Set \( mg = \frac{mv^2}{r} \). The mass \( m \) cancels out from both sides of the equation, simplifying to \( g = \frac{v^2}{r} \).
05

Solve for Maximum Speed

Rearrange the equation \( g = \frac{v^2}{r} \) to solve for \( v \). The maximum speed \( v \) is given by \( v = \sqrt{gr} \), where \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity, and \( r = 45.0 \text{ m} \) is the radius. Substitute these values in to get \( v = \sqrt{9.8 \times 45.0} \approx 21.0 \text{ m/s} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Centripetal Force
When objects are in circular motion, they require a centripetal force to keep moving in that curved path. For our motorcycle traveling over the crest of a hill, the centripetal force is crucial for maintaining its circular path. Without it, the motorcycle would fail to follow the arc of the hill and lose contact with the road.
To calculate this force, we use the formula:
  • \( F_c = \frac{mv^2}{r} \)
Here, \( F_c \) is the centripetal force, \( m \) is the mass of the object (motorcycle in this case), \( v \) is the velocity, and \( r \) is the radius of the circular path.
In simpler terms, the centripetal force depends on how fast the object is moving (velocity), its mass, and the tightness of the curve (radius).
Understanding this force is key to solving problems involving circular motion, like our motorcycle's journey over the hill.
Normal Force
The normal force is a support force acting perpendicular to the surface of contact between two bodies. When the motorcycle travels over the crest of the hill, something interesting happens to this force. Usually, a significant component of normal force acts on objects to counteract gravitational force.
However, at the very top of the hill (the crest), if the motorcycle is going at maximum speed, the normal force is zero. This means gravity alone is directing the motorcycle's path, providing all the necessary force to keep it moving in its circular trajectory.
  • If the motorcycle goes any faster, it would lose contact, as the road can no longer exert enough force to "hold it down."
  • On flat surfaces or less steep areas, the normal force would be significant in maintaining contact with the ground.
Thus, observing the behavior of the normal force at different points of motion is essential for addressing situations involving hills and curves.
Gravitational Force
Gravitational force is the attraction between two masses, like between the Earth and the motorcycle. This force acts downward, pulling the motorcycle toward the Earth. It's calculated using the formula:
  • \( F_g = mg \)
where \( F_g \) is the gravitational force, \( m \) is mass, and \( g \) is the acceleration due to gravity (approx. 9.8 m/s虏).
Over the crest of the hill, the gravitational force becomes more prominent. Since the normal force is zero at maximum speed, the gravitational force is entirely responsible for providing the centripetal force needed to keep the motorcycle on its arcing path.
  • Gravitational force is consistent; it doesn't change based on speed or angle.
  • It powers the centripetal force during the motorcycle's circular motion over the hill.
Recognizing the role of gravitational force helps us understand why the motorcycle can stay on the hillcrest when moving at specific speeds.

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Most popular questions from this chapter

A satellite is in a circular orbit around an unknown planet. The satellite has a speed of \(1.70 \times 10^{4} \mathrm{m} / \mathrm{s},\) and the radius of the orbit is \(5.25 \times 10^{6} \mathrm{m} .\) A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of \(8.60 \times 10^{6} \mathrm{m} .\) What is the orbital speed of the second satellite?

Two satellites, A and B, are in different circular orbits about the earth. The orbital speed of satellite \(\mathrm{A}\) is three times that of satellite \(\mathrm{B}\). Find the ratio \(\left(T_{\mathrm{A}} / T_{\mathrm{B}}\right)\) of the periods of the satellites.

The aorta is a major artery, rising upward from the left ventricle of the heart and curving down to carry blood to the abdomen and lower half of the body. The curved artery can be approximated as a semicircular arch whose diameter is \(5.0 \mathrm{cm} .\) If blood flows through the aortic arch at a speed of \(0.32 \mathrm{m} / \mathrm{s},\) what is the magnitude \(\left(\mathrm{in} \mathrm{m} / \mathrm{s}^{2}\right)\) of the blood's centripetal acceleration?

A special electronic sensor is embedded in the seat of a car that takes riders around a circular loop-the-loop ride at an amusement park. The sensor measures the magnitude of the normal force that the seat exerts on a rider. The loop- the-loop ride is in the vertical plane and its raerts on a rider. The loop- the-loop ride is in the vertical plane and its radius is \(21 \mathrm{m} .\) Sitting on the seat before the ride starts, a rider is level and stationary, and the electronic sensor reads 770 N. At the top of the loop, the rider is upside down and moving, and the sensor reads \(350 \mathrm{N}\). What is the speed of the rider at the top of the loop?

The National Aeronautics and Space Administration (NASA) studies the physiological effects of large accelerations on astronauts. Some of these studies use a machine known as a centrifuge. This machine consists of a long arm, to one end of which is attached a chamber in which the astronaut sits. The other end of the arm is connected to an axis about which the arm and chamber can be rotated. The astronaut moves on a circular path, much like a model airplane flying in a circle on a guideline. The chamber is located \(15 \mathrm{m}\) from the center of the circle. At what speed must the chamber move so that an astronaut is subjected to 7.5 times the acceleration due to gravity?
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