91Ó°ÊÓ

In kinematics, polar coordinates are used to describe the position of a point in a two-dimensional plane, methodically using a radius and angle. This system is particularly useful when dealing with problems involving rotating bodies or systems with radial symmetry, as it allows us to separate the motion into radial and angular components.

To locate a point using polar coordinates, we use two factors:

• The radial distance (r), which tells us how far the point is from a fixed origin.
• The angular position (θ), which measures the angle from a fixed line, often the positive x-axis.

In this exercise, we consider the point B, whose position can be expressed in polar coordinates using the radial distance (abbreviated 'l' for the extension within section OA) and the angle θ about the pivot point O.

Velocity in polar coordinates is a combination of radial and angular components. Let's break it down:

When a point moves, it can experience changes in both its distance from the origin and its angular position. Therefore, the velocity vector can be expressed as:

• The radial velocity, which is the rate of change of the radius with respect to time. Expressed as \( \dot{l} \), it corresponds to how fast point B is moving in the radial direction.
• The angular velocity, expressed as \( (l+7) \cdot \dot{\theta} \), which indicates how fast the point is moving around the pivot in the angular direction.

By calculating the magnitudes and combining these components, we find the velocity of the center B in this exercise to be approximately 2.2006 ft/sec.

Angular motion refers to the change in the orientation of a line segment around a fixed point. When discussing kinematics problems involving rotation, it's essential to understand how angular velocity and angular acceleration contribute.

Here, \( \dot{\theta} = 5 \text{ deg/sec} \) is converted to radians per second as angular velocity. This conversion is necessary because most mathematical models use radians for accuracy and simplicity.

Angular acceleration, \( \ddot{\theta} \), is the rate of change of angular velocity, which in this context is 2 degrees per second squared or approximately 0.0349 rad/sec² after conversion. Knowing these values helps determine how quickly point B changes its angular position as the segment pivots around point O.

Acceleration involves the rate of change of velocity and can also be split into radial and tangential components in polar coordinates.

For radial acceleration, we consider both the rate at which the length changes, \( \ddot{l} \), and the term \(-7\dot{\theta}^2 \), which accounts for changes due to the movement along the arc itself. Tangential acceleration, on the other hand, focuses on the change in angle and how fast that angle is increasing or decreasing, represented by \((l+7)\ddot{\theta} + 2\dot{l}\dot{\theta}\).

The combined effects of these accelerations provide a comprehensive view of how point B's motion evolves, with a computed total acceleration magnitude of approximately 3.8073 ft/sec². An understanding of these two types of acceleration is crucial for analyzing the dynamics of any rotational system.