Kinematics of Particles: Acceleration Calculation in a Slotted Arm Spiral Guide

[amiv4]

Homework Statement

The slotted arm OA forces the small pin to move in the fixed spiral guide defined by r=Kθ. Arm OA starts from rest at θ=π/4 and has a constant counterclockwise angular acceleration $$\ddot{theta}$$ =α. Determine the magnitude of the acceleration of the pin when θ=3π/4.

Answer is a=10.76Kα

any help is appreciated.

The Attempt at a Solution

I have no clue where to start.

 

[nvn]

Hi, amiv4. Start by reading your textbook and studying the example problems. The PF rules state, you must list relevant equations yourself, and show your work; then someone might check your math.

 

[Mene]

it is important to have a thorough understanding of the concepts and principles involved in a problem before attempting to solve it. In this case, the problem involves the kinematics of particles, specifically the acceleration of a pin in a slotted arm spiral guide. To solve this problem, we can use the basic equations of kinematics, which relate the displacement, velocity, and acceleration of an object.

First, we need to understand the given parameters and their relationships. The slotted arm OA forces the small pin to move in a fixed spiral guide, which is defined by the equation r=Kθ. This means that the distance of the pin from the center of the spiral (r) is directly proportional to the angle of rotation (θ) and is determined by the constant K. The arm OA starts from rest at θ=π/4, meaning that the initial velocity of the pin is zero. We are also given that the arm has a constant counterclockwise angular acceleration, \ddot{theta} =α.

To determine the magnitude of the acceleration of the pin when θ=3π/4, we can use the equation for angular acceleration, \ddot{theta} = α. This tells us that the angular acceleration is constant and equal to α throughout the motion of the arm. We can also use the equation for angular displacement, θ=θ0+ω0t+1/2αt^2, where θ0 is the initial angle, ω0 is the initial angular velocity, and t is the time. In this case, we can assume that the initial angular velocity is also zero, as the arm starts from rest. Therefore, the equation simplifies to θ=1/2αt^2.

To find the time it takes for the arm to reach θ=3π/4, we can use the equation for angular velocity, ω=ω0+αt. Since the initial angular velocity is zero, the equation becomes ω=αt. We can rearrange this to solve for t: t=ω/α. Plugging in the values, we get t=3π/8α.

Now, we can use the equation for acceleration, a=dv/dt=ωα, where ω is the angular velocity. Plugging in the values, we get a=3π/8α^2.

Finally, we can use the equation for