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Chapter 3: Problem 24

If the speed of an object in uniform circular motion is constant and the radial distance is doubled, the magnitude of the radial acceleration decreases by what factor? A. 2 B. 3 C. 4 D. 6 E. 1

Short Answer

Expert verified
A. 2

Step by step solution

01

Understand the concept of Radial Acceleration

Radial (or centripetal) acceleration is the rate of change of tangential velocity. The formula for calculating radial acceleration is \(a = v^2/r\), where \(a\) is the radial acceleration, \(v\) is the speed of the object, and \(r\) is the radial distance or radius of the circular path. The radial acceleration always acts towards the centre of the circle.
02

Apply the changes to understand the effect

According to the exercise, the radial distance \(r\) is doubled, i.e., the new distance is \(2r\). We can substitute this new value into the formula. The new acceleration \(a'\) is given by \(a' = v^2/2r\). By comparing with the original formula for acceleration, we notice that the new acceleration is half of the original acceleration.
03

Determine the factor

If the original acceleration is \(a\) and the new acceleration after doubling the radius is \(a'\), we can express the relationship as \(a' = a/2\), which means that the radial acceleration decreases by a factor of 2.

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Key Concepts

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

Centripetal Force
In a circular motion, an object needs a force to keep it moving along a curved path. This force is the centripetal force. It acts towards the center of the circle, constantly altering the direction of the object's velocity without changing its speed.

Key points include:
  • Causation: Centripetal force is not an independent force but is provided by other forces, such as tension, gravity, and friction.
  • Formula: It is calculated as: \[ F_c = \frac{mv^2}{r} \]where \(F_c\) is the centripetal force, \(m\) is the mass of the object, \(v\) is tangential velocity, and \(r\) is the radius.
  • Relationship: If the radius \(r\) is increased, the required centripetal force decreases. This is because a larger circle allows for a gentler curve. But if \(v\) increases, \(F_c\) increases as well, since the object needs more force for the sharper turn.
Uniform Circular Motion
Uniform circular motion refers to motion in a circular path with constant speed. Even though speed is constant, velocity changes since velocity is a vector quantity (involves direction).

Some characteristics include:
  • Constant speed, but changing velocity due to a change in direction.
  • Continuous occurrence of centripetal acceleration, which is directed toward the center of the circle.
  • Balance: In uniform circular motion, the centripetal force is balanced by other forces acting on the object. For instance, planets orbiting the sun utilize gravitational force as the centripetal force.
It helps in understanding natural phenomena such as the Earth's orbit around the Sun.

When studying this type of motion, bear in mind that while speed remains unaltered, the change in velocity points toward a perpetual tug towards the center of the circle where centripetal force plays a crucial part.
Tangential Velocity
Tangential velocity refers to the speed of an object moving along a circular path, and it indicates how fast the object is moving. It is always tangent to the circle at the point of interest, hence its name.

Features of tangential velocity:
  • Direction: It is perpendicular to the radius at any given point of the path.
  • Consistency: In uniform circular motion, the tangential speed remains constant even though the direction of the tangential velocity vector changes.
  • Formula: It is given by:\[ v = \frac{2\pi r}{T} \]where \(v\) is tangential velocity, \(r\) is radius, and \(T\) is the period of the motion.
When the radius is increased, while maintaining the same speed, the path becomes wider, affecting the rate at which the direction changes.

Understanding tangential velocity is crucial for comprehending how objects move along a circular path and how forces interact in such systems.

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Most popular questions from this chapter

The acceleration of a particle in projectile motion A. points along the parabolic path of the particle. B. is directed horizontally. C. vanishes at the particle's highest point. D. is vertically downward. E. is zero. 20\. Adam drops a ball from rest from the top floor of a building at the same time Bob throws a ball horizontally from the same location. Which ball hits the ground first? (Neglect any effects due to air resistance.) A. Adam's ball B. Bob's ball C. They both hit the ground at the same time. D. It depends on how fast Bob throws the ball. E. It depends on how fast the ball falls when Adam drops it.

(a) Explain the difference between an object undergoing uniform circular motion and an object experiencing projectile motion. (b) In what ways are these kinds of motion similar?

Commercial ultracentrifuges can rotate at rates of \(100,000 \mathrm{rpm}\) (revolutions per minute). As a consequence, they can create accelerations on the order of \(800,000 \mathrm{~g}\). (A " \(g\) " represents an acceleration of \(9.8 \mathrm{~m} / \mathrm{s}^{2}\).) Find the distance from the rotation axis of the sample chamber in such a device. Calculate the speed of an object traveling under the given conditions.

Sports In 1993, Javier Sotomayor set a world record of \(2.45 \mathrm{~m}\) in the men's outdoor high jump. He is \(193 \mathrm{~cm}\) ( \(6 \mathrm{ft} 4 \mathrm{in}\).) tall. By treating his body as a point located at half his height, and given that he left the ground a horizontal distance from the bar of \(1.5 \mathrm{~m}\) at a takeoff angle of \(65^{\circ}\), determine Javier Sotomayor's takeoff speed. (Neglect any effects due to air resistance.)

\- You toss a ball into the air at initial angle \(40^{\circ}\) from the horizontal. At what point in the ball's trajectory does the ball have the smallest speed? (Neglect any effects due to air resistance.) A. just after it is tossed B. at the highest point in its flight C. just before it hits the ground D. halfway between the ground and the highest point on the rise portion of the trajectory E. halfway between the ground and the highest point on the fall portion of the trajectory
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