- We already know that the angular acceleration is the rate of change of angular velocity. It is a vector quantity and can be represented both in magnitude and direction with the help of right hand screw rule.
- A body rotating in a plane about a fixed axis is subjected to the angular acceleration. But whenever a rotating body changes its axis of rotation, the body will be subjected to both the angular acceleration and another acceleration which is called as gyroscopic acceleration. Therefore the total acceleration of the rotating body is the sum of angular acceleration of the rotating body and gyroscopic acceleration.

#### Derivation

Consider a disc rotating (spinning) in anticlockwise direction, when seen from the front, about an axis **OX** (known as axis of spin) at an angular speed of **ω**, as shown in Fig (a). Let the axis of spin move to a new position OX’ in short interval of time **δt** such that the angular displacement of the axis of spin is **δθ** and the angular speed of the disc is increased to (**ω+δθ**).

Using the right hand screw rule, the initial angular velocity of the disc (**ω**) is represented by vector **o𝑥** and the final angular velocity of the disc (**ω+δω**) is represented by vector **o𝑥**, as shown in Fig. The vector **𝑥****𝑥’** represents the change of angular velocity in time **δt** i.e., the angular acceleration of the disc.

Resolving vector **𝑥****𝑥’** into two mutually perpendicular components, we get

- Component
**𝑥a**in the direction of**o𝑥**, and - Component
**a𝑥**in the direction perpendicular to**o𝑥**.

##### Component of angular acceleration in the direction of **o𝑥 (α**_{t})

**)**

_{t}**o𝑥**is given by

**α**

_{t}**= dω/d**t represents the change in magnitude of the angular velocity of disc

**‘ω’**with respect to time.

##### Component of angular acceleration in the direction perpendicular to **o𝑥 (****α**_{c}**)**

**o𝑥 (**

_{c}Component of angular acceleration perpendicular to **o𝑥** is given by

Since **δθ** is very small, therefore **sin δθ** = **δθ**, hence we get

In the limit, when **δt** **→0**,

**α _{c} = ω . ω_{t}**

**∵ dθ/dt = ω_{p}**

Thus the expression **α _{c} = ω . ω_{p}** represents the change in direction of the axis of spin with respect to time.

##### Total angular acceleration of the disc (α)

Total angular acceleration of the disc is given by

**α = vector** **𝑥****𝑥** = **α _{t}** +

**α**

_{c}**∴ α = dω/dt + ω .** **ω _{p}**

where **α _{c }**=

**ω .**

**ω**is known as gyroscopic acceleration.

_{p}**ω**=

_{p }**dθ/dt**is known as angular velocity of precession.

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