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Chapter 9: Problem 35

An object moves with a speed \(v\) on a circular path of radius \(r\). If both the speed and the radius are doubled, does the centripetal acceleration of the object increase, decrease, or stay the same? Explain.

Short Answer

Expert verified
The centripetal acceleration doubles.

Step by step solution

01

Understand the Formula for Centripetal Acceleration

The formula for centripetal acceleration \(a_c\) is given by \( a_c = \frac{v^2}{r} \), where \(v\) is the speed of the object and \(r\) is the radius of the circular path.
02

Initial Centripetal Acceleration

Calculate the initial centripetal acceleration with the original speed \(v\) and radius \(r\). Given \( a_{c, ext{initial}} = \frac{v^2}{r} \).
03

Calculate New Velocity and Radius

When both speed and radius are doubled, the new speed \(v' = 2v\) and the new radius \(r' = 2r\).
04

New Centripetal Acceleration

Substitute the new values into the centripetal acceleration formula: \( a_{c, ext{new}} = \frac{(2v)^2}{2r} = \frac{4v^2}{2r} = \frac{2v^2}{r} \).
05

Compare Acceleration Values

Compare \( a_{c, ext{new}} \) with \( a_{c, ext{initial}} \): \( a_{c, ext{new}} = 2 \times a_{c, ext{initial}} \). This means the new acceleration is twice the initial acceleration.

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Key Concepts

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

Circular Motion
Circular motion occurs when an object travels along the circumference of a circle or a circular path. This type of motion is everywhere in our everyday lives, from the orbits of planets to the wheels of your bike.

Here are some key points about circular motion:
  • An object in circular motion continuously changes its direction, even if its speed remains constant. This change in direction is due to the continuous turning of the object.
  • Circular motion can be uniform or non-uniform: uniform when the speed remains constant, and non-uniform when the speed varies.
  • The centripetal force is what keeps an object moving in a circular path, acting towards the center of the circle.
Understanding circular motion is key, especially when considering objects like the one in the original exercise, where the radius and velocity of circular movement interact directly with the object's centripetal acceleration.
Velocity
Velocity is a vector, which means it not only has a magnitude (speed) but also a direction. In the context of circular motion, velocity at any given point is tangential to the path of the motion.

Some crucial points about velocity in circular motion are:
  • Even in uniform circular motion (where speed is constant), velocity is not constant because the direction is continually changing.
  • To change the speed of the object while in circular motion, there would be a radial component of acceleration, affecting the velocity's magnitude.
Velocity is crucial to understanding the concept of centripetal acceleration. In the given exercise, doubling the velocity leads to a significant change in the centripetal acceleration due to its square relationship in the formula.
Radius
Radius is the distance from the center of a circle to any point on its edge. It's a crucial factor that determines many aspects of circular motion, including the path's scale.

Consider the following about radius in circular motion:
  • The radius directly influences the path of motion's length and curvature.
  • In the context of centripetal acceleration, the radius appears in the denominator of the formula. Hence, as the radius changes, it inversely affects the centripetal acceleration.
  • A larger radius means a larger path, resulting in a slower need for acceleration to maintain circular motion at the same speed.
Doubling the radius in the original problem demonstrates this inverse relationship, as it alters not only the scale of motion but also the overall centripetal acceleration dynamics when paired with changes in velocity.

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Most popular questions from this chapter

Skylab, the largest spacecraft ever to fall back to Earth, met its fiery end on July 11, 1979, after flying directly over Everett, Washington, on its last orbit. On the news that night before the crash, broadcaster Walter Cronkite, in his rich baritone voice, said, "NASA says there is \(a\) little chance it will land in a populated area." (Italics added.) After a commercial he immediately corrected himself, saying, "I meant to say 'there is little chance' Skylab will hit a populated area." In fact, it landed primarily in the Indian Ocean off the west coast of Australia, though several pieces were recovered near the town of Esperance, Australia, which sent the U.S. State Department a \(\$ 400\) bill for littering. The cause of Skylab's crash was the friction it experienced in the upper reaches of Earth's atmosphere. As the radius of Skylab's orbit decreased, did its speed increase, decrease, or stay the same?

A driver takes a \(1400-\mathrm{kg}\) car out for a spin, going around a corner with a radius of \(63 \mathrm{~m}\) at a speed of \(18 \mathrm{~m} / \mathrm{s}\). The coefficient of static friction between the car and the road is 0.85. Assuming the car doesn't skid, what is the force exerted on it by static friction?

One day in the future you may take a pleasure cruise to the Moon. While there you might climb a lunar mountain and throw a rock horizontally from its summit. If, in principle, you could throw the rock fast enough, it might end up hitting you in the back. Explain.

When riding in a \(1300-\mathrm{kg}\) car, you go around a curve of radius \(59 \mathrm{~m}\) with a speed of \(16 \mathrm{~m} / \mathrm{s}\). The coefficient of static friction between the car and the road is \(0.88\). Assuming that the car doesn't skid, what is the force exerted on it by static friction?

Think \& Calculate (a) Calculate the orbital period of a satellite that orbits two Earth radii above the surface of Earth. (b) How does your answer to part (a) depend on the mass of the satellite? Explain.
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