Finding Velocity and Acceleration of Different Points in a Solid Body

Understanding the motion of different points in a solid body is crucial in the field of physics, particularly when dealing with rotational motion. When a solid body rotates, each point in the body experiences different velocities and accelerations depending on its position relative to the axis of rotation. In this article, we will explore how to find the velocity and acceleration of different points in a solid body.

Kinematics of Rotational Motion

Before we delve into the specifics, let's review some basic concepts of rotational motion:

• Angular Velocity ($\omega$): The rate of change of angular displacement and is a vector quantity.
• Angular Acceleration ($\alpha$): The rate of change of angular velocity and is also a vector quantity.
• Radius Vector ($r$): The position vector of a point in the body from the axis of rotation.

Velocity in Rotational Motion

The velocity of a point in a rotating body is always perpendicular to the radius vector and lies in the plane of rotation. The magnitude of the velocity ($v$) of a point at a distance $r$ from the axis of rotation is given by the formula:

[ v = r \cdot \omega ]

where:

• $v$ is the linear velocity of the point.
• $r$ is the radius vector (distance from the axis of rotation).
• $\omega$ is the angular velocity.

Acceleration in Rotational Motion

Acceleration in a rotating body can be of two types: centripetal (radial) acceleration and tangential acceleration.

Centripetal (Radial) Acceleration

Centripetal acceleration is directed towards the center of the circular path and is responsible for keeping the point in circular motion. It is given by:

[ a_c = \frac{v^2}{r} = r \cdot \omega^2 ]

where:

• $a_c$ is the centripetal acceleration.
• $v$ is the linear velocity of the point.
• $r$ is the radius vector.
• $\omega$ is the angular velocity.

Tangential Acceleration

Tangential acceleration is due to the change in the magnitude of the velocity and is directed along the tangent to the circular path. It is given by:

[ a_t = r \cdot \alpha ]

where:

• $a_t$ is the tangential acceleration.
• $r$ is the radius vector.
• $\alpha$ is the angular acceleration.

Total Acceleration

The total acceleration ($a$) of a point in a rotating body is the vector sum of centripetal and tangential accelerations:

[ a = \sqrt{a_c^2 + a_t^2} ]

Differences and Important Points

Here is a table summarizing the differences between velocity and acceleration in rotational motion:

Property	Velocity ($v$)	Acceleration ($a$)
Direction	Perpendicular to radius vector $r$	$a_c$: Towards center, $a_t$: Tangent to path
Formula	$v = r \cdot \omega$	$a_c = r \cdot \omega^2$, $a_t = r \cdot \alpha$
Depends on	Radius $r$ and angular velocity $\omega$	Radius $r$, angular velocity $\omega$, and angular acceleration $\alpha$
Units	Meters per second (m/s)	Meters per second squared (m/s²)

Examples

Example 1: Velocity of a Point on a Rotating Wheel

Consider a point on the edge of a wheel that is rotating with an angular velocity of 10 rad/s. If the radius of the wheel is 0.5 m, the velocity of the point is:

[ v = r \cdot \omega = 0.5 \, \text{m} \times 10 \, \text{rad/s} = 5 \, \text{m/s} ]

Example 2: Acceleration of a Point on a Rotating Disk

A disk is rotating with an angular velocity of 20 rad/s and an angular acceleration of 2 rad/s². A point on the disk is 0.3 m from the center. The centripetal and tangential accelerations are:

[ a_c = r \cdot \omega^2 = 0.3 \, \text{m} \times (20 \, \text{rad/s})^2 = 120 \, \text{m/s}^2 ] [ a_t = r \cdot \alpha = 0.3 \, \text{m} \times 2 \, \text{rad/s}^2 = 0.6 \, \text{m/s}^2 ]

The total acceleration is:

[ a = \sqrt{a_c^2 + a_t^2} = \sqrt{120^2 + 0.6^2} \approx 120.01 \, \text{m/s}^2 ]

In conclusion, finding the velocity and acceleration of different points in a solid body involves understanding the relationship between linear and angular quantities. The formulas provided are essential for solving problems in rotational motion, and the examples illustrate how to apply these concepts in practical scenarios.