91Ó°ÊÓ

A 1 -kg collar can slide on a horizontal rod that is free to rotate about a vertical shaft. The collar is initially held at \(A\) by a cord attached to the shaft. A spring of constant \(30 \mathrm{N} / \mathrm{m}\) is attached to the collar and to the shaft and is undeformed when the collar is at \(A .\) As the rod rotates at the rate \(\dot{\theta}=16 \mathrm{rad} / \mathrm{s}\), the cord is cut and the collar moves out along the rod. Neglecting friction and the mass of the rod, determine (a) the radial and transverse components of the acceleration of the collar at \(A,(b)\) the acceleration of the collar relative to the rod at \(A,(c)\) the transverse component of the velocity of the collar at \(B .\)

Step by step solution

01

Identify the Given Data

We have a collar of mass 1 kg, a spring constant of 30 N/m, and the initial angular velocity of the rod is \( \dot{\theta} = 16 \mathrm{rad}/\mathrm{s} \). The spring is undeformed initially, and we are analyzing the situation immediately after the cord is cut.

02

Determine Radial Acceleration at A

Component (a): The radial acceleration is due to the spring force. Since the spring is initially undeformed at A, the initial radial acceleration is zero because there is no displacement from the equilibrium position of the spring.

03

Calculate Transverse Acceleration at A

Component (b): The transverse or angular acceleration at A is zero initially because \( \dot{\theta} = 16 \mathrm{rad}/\mathrm{s} \) is constant.

04

Compute Acceleration Relative to the Rod at A

Component (c): Since the spring and the rod have no initial displacement, the only motion after the cord is cut is due to the rotational speed. The acceleration relative to the rod is primarily the centripetal acceleration, \( a_c = r \dot{\theta}^2 \). As the spring is initially undeformed, the initial relative radial acceleration is zero.

05

Find the Transverse Velocity at B

Since transverse velocity depends solely on the angular velocity of the collar, which does not change as the collar moves along the rod (since \( \dot{\theta} \) is constant), the transverse component of velocity remains \( v_\theta = r \cdot \dot{\theta} \). Here \( r \) at B needs to be calculated based on the equilibrium spring extension.

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Key Concepts

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

Kinematics

Kinematics involves the description of motion, focusing on the positions, velocities, and accelerations of objects without considering the forces that cause these movements. In this exercise, the horizontal rod rotates about a vertical shaft, affecting the collar's motion along the rod as the cord is cut. The kinematic aspects are described by:

• Angular position and velocity: The angle \( \theta \) and angular velocity \( \dot{\theta} = 16 \text{ rad/s} \).
• Radial and transverse components: Radial motion occurs along the rod, while transverse motion occurs perpendicular to the radial direction due to rotation.

Understanding these, we analyze the collar's movement relative to its initial state and how it changes over time.

Centripetal Acceleration

Centripetal acceleration always acts perpendicular to the velocity of an object moving along a curved path, pointing towards the center of rotation. This specific type of acceleration keeps the object in a circular motion path. In our problem, the collar experiences centripetal acceleration because:

• The rod rotates at a constant angular velocity, \( \dot{\theta} \).
• The radial distance changes as the collar moves outwards.

Thus, the centripetal acceleration can be calculated using \( a_c = r \dot{\theta}^2 \), where \( r \) is the radius from the axis of rotation to the collar's current position. At point A, right after the cord is severed, this directly influences how the collar behaves.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves around an axis. It is expressed in radians per second in the context of this exercise. For the rotating rod and the sliding collar, the angular velocity \( \dot{\theta} = 16 \text{ rad/s} \) remains constant, meaning:

• The collar's transverse component of velocity is determined purely by this constant rate.
• The radial component does not affect the rod's angular velocity, but influences the collar's position.

This constant rotation enables us to predict how the motion's angular aspect contributes to the overall dynamics of the system.

Radial and Transverse Components of Acceleration

When dealing with rotational systems, acceleration can be broken down into radial and transverse components. These are crucial because:

• Radial acceleration accounts for changes in distance from the rotation axis, often influenced by forces like the spring.
• Transverse acceleration, being perpendicular to radial acceleration, does not occur when rotational speed is constant.

For this scenario:

• The collar initially has zero radial acceleration at point A because of no displacement caused by an undeformed spring.
• Transverse acceleration is likewise zero since the constant angular velocity ensures no change in rotational rate.

Clearly distinguishing these components helps us thoroughly understand how the collar accelerates under these specific conditions.