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Chapter 2: Problem 4

Circling particle and force Two particles of mass \(m\) and \(M\) undergo uniform circular motion about each other at a separation \(R\) under the influence of an attractive constant force \(F\). The angular velocity is \(\omega\) radians per second. Show that \(R=\left(F / \omega^{2}\right)(1 / m+1 / M)\).

Short Answer

Expert verified
The separation distance is \( R = \frac{F}{\omega^2} \left( \frac{1}{m} + \frac{1}{M} \right) \).

Step by step solution

01

Understanding the Forces Involved

To solve this problem, we begin by understanding that the force causing the circular motion is the same force, an attractive constant force, acting between the two masses, given as \( F \). This force should equal the centripetal force required for each of the masses to perform circular motion.
02

Apply Centripetal Force Formula

For an object of mass \( m \), the required centripetal force to maintain circular motion with angular velocity \( \omega \) and radius of motion \( R_m \) is \( mR_m \omega^2 \). For an object of mass \( M \), it becomes \( MR_M \omega^2 \). The sum of these radii must equal \( R \), i.e., \( R_m + R_M = R \).
03

Equating Forces

The attractive constant force \( F \) should provide the necessary centripetal force, which means that \( F = mR_m \omega^2 = MR_M \omega^2 \). By rearranging, \( R_m = \frac{F}{m \omega^2} \) and \( R_M = \frac{F}{M \omega^2} \).
04

Calculate Total Separation Distance

The total separation distance \( R \) between the two particles is the sum of the radii \( R = R_m + R_M \). Using the expressions for \( R_m \) and \( R_M \), we have \( R = \frac{F}{m \omega^2} + \frac{F}{M \omega^2} \).
05

Simplify the Expression

Factor out \( \frac{F}{\omega^2} \) from the expression: \[ R = \frac{F}{\omega^2} \left( \frac{1}{m} + \frac{1}{M} \right) \]This matches the expression we were asked to show.

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Key Concepts

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

Centripetal Force
When we hear about circular motion, centrifrugal force often comes into play. In uniform circular motion, an object continually changes its direction, moving along a circular path. To keep an object following this path, it needs a net inward force, which is what we call the centripetal force. This force is always directed towards the center of the circle and is crucial for maintaining the object's path.

For an object with mass \( m \) moving in a circle with radius \( R \) and angular velocity \( \omega \), the required centripetal force is given by the formula \[ F_{c} = mR\omega^2 \]It essentially tells us that the greater the mass of the object or the faster it rotates, the more force is necessary to keep it moving in a circular path.

Let's remember that this is not a new type of force but one or several of the existing forces (like gravity or tension) playing the role of pulling the object inward.
Attractive Constant Force
When dealing with two particles performing circular motion around each other, there must be some force attracting them towards each other. This is termed as an attractive constant force in this context. This force is crucial for sustaining the circular motion and is constant in magnitude across the motion of the particles.

In our scenario, both particles exert and experience this force, \( F \), which effectively becomes the centripetal force required by each particle to stay in its orbit. Therefore, this attractive force is responsible for controlling and limiting the separation between them as they rotate.

Such forces could be gravitational forces or electromagnetic forces depending on the context, but they share the need to be present constantly and equally to keep the system stable.
Angular Velocity
Angular velocity is a measure that describes how quickly an object is rotating. It is basically the rate at which an object moves around a circle and is expressed in radians per second. This is a crucial factor because it directly influences the force needed to maintain circular motion.

Mathematically, if an object moves in a circle of radius \( R \), then its angular displacement \( \theta \) after time \( t \) is given by \[ \theta = \omega \cdot t \]Here, \( \omega \) is the angular velocity.

A higher angular velocity implies a faster rotation, which demands a higher centripetal force to keep the object on its circular path. It's fascinating how deeply interwoven these concepts are, as changing one of them (like increasing the angular velocity) directly impacts others (necessitating more centripetal force).

Understanding these interconnections helps in analyzing systems involving rotational dynamics, where objects are not just moving but moving in a circle consistently.

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Most popular questions from this chapter

Mass in cone A particle of mass \(m\) slides without friction on the inside of a cone. The axis of the cone is vertical, and gravity is directed downward. The apex half-angle of the cone is \(\theta\), as shown. The path of the particle happens to be a circle in a horizontal plane. The speed of the particle is \(v_{0} .\) Draw a force diagram and find the radius of the circular path in terms of \(v_{0}, g\), and \(\theta\).

Leaning pole A pole of negligible mass leans against a wall, at angle \(\theta\) with the horizontal. Gravity is directed down. (a) Find the constraint relating the vertical acceleration of one end to the horizontal acceleration of the other. (b) Now suppose that each end carries a pivoted mass \(M\). Find the initial vertical and horizontal components of acceleration as the pole just begins to slide on the frictionless wall and floor. Assume that at the beginning of the motion the forces exerted by the rod are along the line of the rod. (As the motion progresses, the system rotates and the rod exerts sidewise forces.)

Concrete mixer' In a concrete mixer, cement, gravel, and water are mixed by tumbling action in a slowly rotating drum. If the drum spins too fast the ingredients stick to the drum wall instead of mixing. Assume that the drum of a mixer has radius \(R=0.5 \mathrm{~m}\) and that it is mounted with its axle horizontal. What is the fastest the drum can rotate without the ingredients sticking to the wall all the time? Assume \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\)

Planck units \(^{*}\) Max Planck introduced a constant \(h\), now called Planck's constant, to relate the energy of an oscillator to its frequency. \(h=6.6 \times\) \(10^{-34} \mathrm{~J} \cdot \mathrm{s}\), where 1 joule \((\mathrm{J})=1\) newton-meter. \((h\) is engraved on Planck's tombstone in G枚ttingen, Germany.) Planck pointed out that if one takes \(h\) and Newton's gravitational constant \(G=6.7 \times 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) and the speed of light \(c=3.0 \times\) \(10^{8} \mathrm{~m} / \mathrm{s}\) as fundamental quantities, it is possible to combine them to form three new independent quantities to replace the customary units of mass, length, and time. The three new quantities are called the Planck units. (a) Planck length \(L_{p}\) (b) Planck mass \(M_{p}\) (c) Planck time \(T_{p}\). The Planck units have a natural role in modern cosmology, particularly the cosmology of the early universe. Find the SI values of the Planck units, as for example \(1 L_{p}=\) (?) \(\mathrm{m}\). (Note: published results may differ from yours because they are often evaluated using \(\hbar=h / 2 \pi .)\)

Time-dependent force \(^{*}\) A \(5-\mathrm{kg}\) mass moves under the influence of a force \(\mathbf{F}=\left(4 t^{2} \hat{\mathbf{i}}-3 t \hat{\mathbf{j}}\right) \mathrm{N}\), where \(t\) is the time in seconds \((1 \mathrm{~N}=1\) newton \() .\) It starts at rest from the origin at \(t=0\). Find: \((a)\) its velocity; \((b)\) its position; and (c) \(\mathbf{r} \times \mathbf{v}\), for any later time.
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