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Chapter 1: Problem 3

An object is resting on a platform that rotates at a constant speed. At first, it is a distance of half the platform鈥檚 radius from the center. If the object is moved to the edge of the platform, what happens to the centripetal force that it experiences? Assume the platform continues rotating at the same speed. (A) Increases by a factor of 4 (B) Increases by a factor of 2 (C) Decreases by a factor of 2 (D) Decreases by a factor of 4

Short Answer

Expert verified
(C) Decreases by a factor of 2

Step by step solution

01

Understanding the Centripetal Force formula

Centripetal force is given by the formula \(F = m \cdot v^{2} / r \), where \( F \) is the centripetal force, \( m \) is the mass of the object, \( v \) is the speed of the object and \( r \) is the radius or distance of the object from the center of the circle. In this problem, the mass and speed of the object remain constant, but the radius changes.
02

Calculating Centripetal Force at Different Radii

Initially, the object is a distance of half the platform鈥檚 radius from the center. Let's say the initial radius is \( r \), so the initial centripetal force \( F_{i} \) is \( F_{i} = m \cdot v^{2} / r \). Then the object is moved to the edge of the platform, which means the new radius is \( 2r \). So, the new centripetal force \( F_{f} \) is \( F_{f} = m \cdot v^{2} / 2r \).
03

Comparing the Initial and Final Centripetal Forces

To find out how much the centripetal force changes when the object moves from a distance of half the platform鈥檚 radius to the edge, we will divide the final centripetal force \( F_{f} \) by the initial centripetal force \( F_{i} \). This gives us \( F_{f} / F_{i} = (m \cdot v^{2} / 2r) / (m \cdot v^{2} / r) = 1/2 \), which means that the centripetal force 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.

Circular Motion
Circular motion is a fundamental concept in physics that deals with the movement of objects along a circular path. When an object moves in a circle, it continuously changes direction, even if its speed remains constant.
This constant change in direction requires a force to act on the object, known as centripetal force. Centripetal means 'center-seeking,' referring to the fact that this force always points towards the center of the circle.
It keeps the object moving along its curved path and prevents it from flying off tangentially.
  • Key Elements of Circular Motion:
    • Uniform circular motion occurs when an object travels around the circle at a constant speed.
    • The velocity of the object changes because its direction changes, even if speed remains unchanged.
    • The centripetal force doesn't change the object's speed, only its direction.
Understanding these concepts can help in solving problems related to objects moving in circles, as seen with the rotating platform problem.
Rotational Dynamics
Rotational dynamics is the branch of physics that deals with the effect of forces on rotational motion. Unlike linear motion, where objects move in straight lines, rotational dynamics focuses on the circular paths and the forces involved. An essential aspect of rotational dynamics in circular motion is understanding how forces like centripetal force act.
This force allows objects to follow a circular path by constantly changing their direction.
  • Key Points in Rotational Dynamics:
    • Torque is the rotational equivalent of force. It acts to change an object's rotational motion.
    • Moment of inertia is a measure of an object's resistance to changes in rotation, much like mass in linear motion.
    • Centripetal force is crucial in keeping objects on their circular path without changing their speed.
      • By understanding rotational dynamics, students can better comprehend how forces and motion work together in systems involving rotation, like our platform scenario.
Physics Problem Solving
When solving physics problems, especially those involving circular motion or rotational dynamics, having a structured approach is beneficial.To solve the problem about the object on the rotating platform, begin by identifying known variables and relationships:
  • Recognize that mass and speed remain constant.
  • The centripetal force formula: \(F = m \cdot v^{2} / r\)
Next, relate changes to the known variables:
  • Identify initial and final conditions for variables like radius.
  • Apply the formula to determine how changes affect the force.
Finally, verify outcomes:
  • Check calculations to ensure they align with physical principles.
  • Compare outcomes to expected results, like the force decreasing when moving to the platform鈥檚 edge.
Effective problem solving in physics involves integrating these steps to understand the problem comprehensively and arrive at the correct solution.

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Most popular questions from this chapter

A block of mass \(M\) is at rest on a table. It is connected by a string and pulley system to a block of mass \(m\) hanging off the edge of the table. Assume the hanging mass is heavy enough to make the resting block move. If the acceleration of the system and the masses of the blocks are known, which of the following could NOT be calculated? (A) Net force on each block (B) Tension in the string (C) Coefficient of kinetic friction between the table and the block of (D) The speed of the block of mass \(M\) when it reaches the edge of the table

Both of the above strings have their ends locked in place. The first string, \(S_{1},\) is twice as long as the second string, \(S_{2}\) . If sound waves are going to be sent through both, what is the correct ratio of the fundamental frequency of \(S_{1}\) the fundamental frequency of \(S_{2} ?\) (A) \(2 : 1\) (B) \(\sqrt{2} : 1\) (B) \(\sqrt{2} : 1\) (C) \(1 : \sqrt{2}\) (D) \(1 : 2\)

Two spheres of equal size and equal mass are rotated with an equal amount of torque. One of the spheres is solid with its mass evenly distributed throughout its volume, and the other is hollow with all of its mass concentrated at the edges. Which sphere would rotate faster? (A) Solid sphere (B) Hollow sphere (C) They would rotate at equal rates. (D) Additional information is required to determine the relative rates of rotation.

A piano tuner needs to double the frequency of note that a particular string is playing. Should he/she tighten or loosen the string, and by what factor? (A) Tighten, 4 (B) Tighten, 2 (C) Loosen, 2 (D) Loosen, 4

An ambulance is driving toward you. As it approaches, which of the following correctly describe the changes in the sound of the siren鈥檚 pitch and intensity? (A) Increasing pitch, increasing intensity (B) Increasing pitch, decreasing intensity (C) Decreasing pitch, increasing intensity (D) Decreasing pitch, decreasing intensity
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