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Chapter 5: Problem 112

A particle is moving along a circular path. The angular velocity, linear velocity, angular acceleration, and centripetal acceleration of the particle at any instant, respec-, tively, are \(\vec{\omega}, \vec{v}, \vec{\alpha}\), and \(\vec{a}_{c} .\) Which of the following relations is not correct? a. \(\vec{\omega} \perp \vec{v}\) b. \(\vec{\omega} \perp \vec{\alpha}\) c. \(\vec{\omega} \perp \vec{a}_{c}\) d. \(\vec{v} \perp \vec{a}_{c}\)

Short Answer

Expert verified
Relation b, \( \vec{\omega} \perp \vec{\alpha} \), is not correct.

Step by step solution

01

Analyze Relationship Between Angular Velocity and Linear Velocity

Angular velocity \( \vec{\omega} \) is perpendicular to the linear velocity \( \vec{v} \) in circular motion. The linear velocity is tangent to the circle, while angular velocity is along the rotational axis, so this condition \( \vec{\omega} \perp \vec{v} \) is true.
02

Consider Angular Velocity and Angular Acceleration

While both angular velocity \( \vec{\omega} \) and angular acceleration \( \vec{\alpha} \) are vectors along the axis of rotation, they actually may not be perpendicular to each other. They are typically co-linear in linear motion, therefore \( \vec{\omega} \perp \vec{\alpha} \) is not true.
03

Examine Angular Velocity and Centripetal Acceleration

Angular velocity \( \vec{\omega} \) is along the axis of rotation, and centripetal acceleration \( \vec{a}_{c} \) is directed towards the center of the circle along the radius. These vectors are perpendicular, confirming \( \vec{\omega} \perp \vec{a}_{c} \). Thus, this is true.
04

Analyze Linear Velocity and Centripetal Acceleration

Linear velocity \( \vec{v} \) is tangent to the circle, while centripetal acceleration \( \vec{a}_{c} \) points towards the center. These vectors are indeed perpendicular, so the statement \( \vec{v} \perp \vec{a}_{c} \) holds true.

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Key Concepts

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

Angular Velocity
Angular velocity, represented by the vector \( \vec{\omega} \), describes how fast an object is rotating along its axis. It's a measure of the rate of rotation and is usually expressed in radians per second. In the context of circular motion, the angular velocity is crucial because it determines how quickly the object completes a circle or any part of a circular path.
  • Angular velocity is a vector quantity, meaning it has both magnitude and direction.
  • The direction of \( \vec{\omega} \) is determined by the right-hand rule: curl the fingers of your right hand in the direction of rotation, and your thumb will point in the direction of the angular velocity vector.
Because angular velocity is perpendicular to the linear velocity vector \( \vec{v} \), it ensures that the linear velocity is tangent to the circular path while \( \vec{\omega} \) points along the axis of rotation.
Linear Velocity
Linear velocity \( \vec{v} \) in circular motion refers to the speed at which an object travels along the circular path. Unlike angular velocity, linear velocity is concerned with the path's circumference. It is represented as a tangent to the circle at any point of motion.
  • The magnitude of linear velocity can be calculated using the formula \( v = r \cdot \omega \), where \( r \) is the radius of the circle.
  • Linear velocity is directly proportional to both the radius of the path and the angular velocity.
In circular motion, linear velocity is always perpendicular to the centripetal acceleration \( \vec{a}_{c} \), reinforcing the object鈥檚 movement along the tangent without deviating from the circular path.
Centripetal Acceleration
Centripetal acceleration \( \vec{a}_{c} \) is what keeps an object moving in a circular path. It points towards the center of the circle and is responsible for the continuous change in the direction of the linear velocity vector, even if its speed remains constant.
  • The formula for centripetal acceleration is \( a_c = \frac{v^2}{r} \), highlighting that it is proportional to the square of the linear velocity and inversely proportional to the radius.
  • Even when the speed of an object remains constant, the presence of centripetal acceleration causes a constant change in direction, maintaining circular motion.
The perpendicular relationship between linear velocity \( \vec{v} \) and centripetal acceleration \( \vec{a}_{c} \) ensures that while \( \vec{a}_{c} \) pulls the object toward the center, \( \vec{v} \) allows it to continue traveling along the tangent of the circle.
Angular Acceleration
Angular acceleration \( \vec{\alpha} \) describes the rate of change of angular velocity. It provides insight into whether an object is speeding up or slowing down its rate of rotation. Like angular velocity, angular acceleration is also a vector and typically shares the same axis of rotation.
  • When an object's rate of spin increases, angular acceleration is positive; if it decreases, angular acceleration is negative.
  • The units of angular acceleration are typically radians per second squared \( \, \text{rad/s}^2 \).
In certain situations, angular velocity and angular acceleration may be co-linear, especially when no external torques cause changes, leading them to point in the same direction. This can make them not perpendicular as indicated by rotational symmetry in the absence of disturbances.

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Most popular questions from this chapter

The equation of motion of a projectile is $$ y=12 x-\frac{3}{4} x^{2} $$ The horizontal component of velocity is \(3 \mathrm{~ms}^{-1}\). What is the range of the projectile? a. \(18 \mathrm{~m}\) b. \(16 \mathrm{~m}\) c. \(12 \mathrm{~m}\) d. \(21.6 \mathrm{~m}\)

A plank is moving on a ground with a velocity \(v\) and a block is moving on the plank with a velocity \(u\) as shown in figure. What is the velocity of block with respect to ground? a. \(v-u\) towards right b. \(v-u\) towards left c. \(u\) towards right d. None of these

A golfer standing on level ground hits a ball with a velocity of \(u=52 \mathrm{~m} / \mathrm{s}\) at an angle \(a\) above the horizontal. If \(\tan a=5 / 12\), then the time for which the ball is at least 15 \(\mathrm{m}\) above the ground will be (take \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) ) a. \(1 \mathrm{~s}\) b. \(2 \mathrm{~s}\) c. \(3 \mathrm{~s}\) d. \(4 \mathrm{~s}\)

A rifle shoots a bultet with a muzzle velocity of \(400 \mathrm{~m} / \mathrm{s}\) at a small target \(400 \mathrm{~m}\) away. The height above the target at which the bullet must be aimed to hit the target is \(\left(g=10 \mathrm{~ms}^{-2}\right)\) a. \(1 \mathrm{~m}\) b. \(5 \mathrm{~m}\) c. \(10 \mathrm{~m}\) d. \(0.5 \mathrm{~m}\)

The speed of a projectile at its maximum height is \(\sqrt{3} / 2\) times its initial speed. If the range of the projectile is \(P\) times the maximum height attained by it, \(P\) is equal to a. \(4 / 3\) b. \(2 \sqrt{3}\) c. \(4 \sqrt{3}\) d. \(3 / 4\)
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