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Chapter 11: Problem 63

(III) An ant crawls with constant speed outward along a radial spoke of a wheel rotating at constant angular velocity \(\omega\) about a vertical axis. Write a vector equation for all the forces (including inertial forces) acting on the ant. Take the \(x\) axis along the spoke, \(y\) perpendicular to the spoke pointing to the ant's left, and the \(z\) axis vertically upward. The wheel rotates counterclockwise as seen from above.

Short Answer

Expert verified
The ant experiences centrifugal force outward and Coriolis force to its left.

Step by step solution

01

Identify the forces acting on the ant

The ant experiences multiple forces: its weight due to gravity, the normal force from the wheel, and an inertial force due to the wheel's rotation. The gravitational force acts vertically downward, while the normal force acts perpendicular to the wheel's surface. The rotation of the wheel gives rise to a centrifugal force and a Coriolis force, both of which are inertial forces.
02

Express the gravitational force vector

The weight of the ant can be expressed as a vector in the downward direction along the z-axis. This force is given by \(\vec{F}_g = -mg \hat{k}\), where \(m\) is the mass of the ant, and \(\hat{k}\) is the unit vector in the z-direction.
03

Express the normal force vector

The normal force exerted by the wheel is opposite in direction to the gravitational force component acting perpendicular to the wheel's surface. If we assume no vertical acceleration of the ant, this force balances the z-component of the gravitational force, so \(\vec{F}_n = mg \hat{k}\).
04

Determine the centrifugal force vector

For an object in a rotating reference frame, the centrifugal force acts outward along the radius. This force can be expressed as \(\vec{F}_{cf} = mR\omega^2 \hat{i}\), where \(R\) is the radial distance from the axis of rotation to the ant and \(\hat{i}\) is the unit vector along the spoke (x-axis).
05

Determine the Coriolis force vector

The Coriolis force arises due to the ant's radial velocity in the rotating frame and acts perpendicular to the velocity vector in the direction of \(\hat{j}\). The magnitude is given by \(\vec{F}_{cor} = -2m\omega v_r \hat{j}\), where \(v_r\) is the radial speed of the ant. The negative sign is due to the rotation direction.
06

Write the complete force vector equation

Sum up all the forces to get the net force acting on the ant. This gives: \[ \vec{F}_{net} = \, \vec{F}_g + \vec{F}_n + \vec{F}_{cf} + \vec{F}_{cor} = (-mg \hat{k}) + (mg \hat{k}) + (mR\omega^2 \hat{i}) + (-2m\omega v_r \hat{j}) \] Simplifying yields \[ \vec{F}_{net} = mR\omega^2 \hat{i} - 2m\omega v_r \hat{j} \].

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Key Concepts

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

Centrifugal Force
When an object is in a rotating system, like our ant on the spinning wheel, it experiences a force that seems to push it outward. This is known as centrifugal force. It鈥檚 not a real force but rather an effect due to the object's inertia in the rotating frame.
Centrifugal force acts along the radial direction from the center of rotation, meaning it pushes the ant outward from the wheel's axis.
  • Formula: \( \vec{F}_{cf} = mR\omega^2 \hat{i} \)
  • Where \( m \) is the mass, \( R \) is the radial distance, and \( \omega \) is the angular velocity.
  • The unit vector \( \hat{i} \) indicates the direction along the spoke (x-axis).
So, if the wheel spins faster or the ant moves further out, the centrifugal effect increases. It's purely a rotational result and doesn鈥檛 exist in non-rotating frames.
Coriolis Force
Coriolis force arises in rotating systems and affects objects moving within those systems. For our ant, this force appears due to its motion along the wheel鈥檚 spoke while the wheel rotates.
  • It acts perpendicular to the ant's velocity vector and can change its direction, but not speed.
  • Formula: \( \vec{F}_{cor} = -2m\omega v_r \hat{j} \)
  • Where \( m \) is the mass of the ant, \( \omega \) is the angular velocity, and \( v_r \) is the radial speed of the ant.
  • The negative sign accounts for the counterclockwise rotation direction, while \( \hat{j} \) shows the direction perpendicular to the spoke (y-axis).
The Coriolis effect is crucial in understanding the rotation-induced forces moving across the radial direction. It鈥檚 a result of the Earth's rotation and is significant in meteorology.
Inertial Forces
Inertial forces arise when observing motion in non-inertial (accelerating or rotating) frames of reference. Our ant on the rotating wheel experiences centrifugal and Coriolis forces, which are inertial forces.
  • These forces do not act on the object in an inertial (non-accelerating) frame but appear due to the acceleration of the reference frame itself.
  • In a rotating frame, inertial forces contribute to perceived accelerations and movement directions for objects within that system.
In the context of rotational dynamics, inertial forces provide insights into the object's motion and the effects of rotation. Understanding these forces is essential for describing dynamics in rotating systems like planets, weather patterns, and mechanical systems.

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Most popular questions from this chapter

A nonrotating cylindrical disk of moment of inertia \(I\) is dropped onto an identical disk rotating at angular speed \(\omega\). Assuming no external torques, what is the final common angular speed of the two disks?

Water drives a waterwheel (or turbine) of radius \(R=3.0 \mathrm{~m}\) as shown in Fig. \(11-47 .\) The water enters at a speed \(v_{1}=7.0 \mathrm{~m} / \mathrm{s}\) and exits from the waterwheel at a speed \(v_{2}=3.8 \mathrm{~m} / \mathrm{s} . \quad(a)\) If \(85 \mathrm{~kg}\) of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? \((b)\) What is the torque the water applies to the waterwheel? \((c)\) If the water causes the waterwheel to make one revolution every \(5.5 \mathrm{~s}\), how much power is delivered to the wheel?

(II) The origin of a coordinate system is at the center of a whecl which rotates in the \(x y\) plane about its axle which is the \(z\) axis. A force \(F=215 \mathrm{N}\) acts in the \(x y\) plane, at a \(+33.0^{\circ}\) angle to the \(x\) axis at the point \(x=28.0 \mathrm{cm}, y=33.5 \mathrm{cm}\) . Determine the magnitude and direction of the torque produced by this force about the axis.

A toy gyroscope consists of a 170 -g disk with a radius of \(5.5 \mathrm{~cm}\) mounted at the center of a thin axle \(21 \mathrm{~cm}\) long (Fig. 11-41). The gyroscope spins at \(45 \mathrm{rev} / \mathrm{s}\). One end of its axle rests on a stand and the other end precesses horizontally about the stand. (a) How long does it take the gyroscope to precess once around? \((b)\) If all the dimensions of the gyroscope were doubled (radius \(=11 \mathrm{~cm},\) axle \(=42 \mathrm{~cm})\) how long would it take to precess once?

A 220-g top spinning at 15 rev/s makes an angle of \(25^{\circ}\) to the vertical and precesses at a rate of 1.00 rev per \(6.5 \mathrm{~s}\). If its \(\mathrm{CM}\) is \(3.5 \mathrm{~cm}\) from its tip along its symmetry axis, what is the moment of inertia of the top?
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