Short Answer:

Combined motion is the type of motion in which a body experiences both translation and rotation at the same time. It means that while the body moves forward in a straight or curved path (translation), it also spins or rotates about its own axis (rotation).

A common example of combined motion is the rolling motion of a wheel on a road. In this motion, the wheel translates forward while simultaneously rotating about its center. Most machine parts, such as gears, pulleys, and vehicle wheels, exhibit combined motion during operation.

Detailed Explanation :

Combined Motion

Combined motion is a general form of motion in which a body undergoes both translational motion and rotational motion simultaneously. It is also known as general plane motion because the body moves in such a way that every point on it describes a path lying in a single plane, but not all points move in the same direction or at the same speed.

In simpler terms, when a rigid body moves forward (translation) and at the same time spins or rotates (rotation), the motion is called combined motion. This type of motion is common in mechanical systems such as wheels, gears, cams, and rolling cylinders. The study of combined motion helps engineers understand the behavior of moving parts in machines and design efficient mechanisms for motion transmission.

Definition

Combined motion can be defined as:

“The motion of a rigid body in which both translation and rotation occur simultaneously is called combined motion.”

It can also be described as the combination of linear and angular motion of a body in a single plane.

Mathematically, if a rigid body has both linear velocity and angular velocity , then the velocity of any point on the body is given by:

where,
= velocity of point A,
= velocity of the center of mass or reference point,
= angular velocity,
= distance of point A from the axis of rotation.

This shows that the motion of a point on the body depends on both translational and rotational components.

Explanation of Combined Motion

To understand combined motion, consider a wheel rolling on a flat surface.

The center of the wheel moves forward in a straight line — this represents translation.
At the same time, the wheel rotates about its own axis — this represents rotation.
Hence, the wheel undergoes combined (translational + rotational) motion.

In this type of motion, different points on the wheel have different velocities. The top point moves at twice the speed of the center, the center moves with the velocity of translation, and the bottom point (point of contact) has zero velocity because it is momentarily at rest. This behavior is the best example of combined motion in engineering mechanics.

Types of Combined Motion

Rolling Motion:
It is the most common type of combined motion where a circular body, such as a wheel or ball, rolls over a surface.
Example: Motion of a bicycle wheel or a rolling cylinder.
Slipping and Rolling Motion:
When a rolling body also slides due to friction or lack of grip, both sliding (translation) and rotation occur.
Example: A car wheel skidding on a slippery road.
Swinging Motion:
A pendulum exhibits combined motion because it both translates and rotates about its point of suspension.

Thus, combined motion can occur in different forms depending on the shape of the body and the forces acting on it.

Velocity in Combined Motion

In combined motion, the velocity of any point on the body is the vector sum of its translational velocity and rotational velocity.

Let the center of a rolling wheel move with a linear velocity , and the wheel rotates with angular velocity .
Then,

where is the radius of the wheel.

The top point of the wheel has velocity .
The bottom point (in contact with ground) has velocity .
The center moves with velocity .

This relationship clearly shows the combined nature of motion, as both translation and rotation are involved.

Acceleration in Combined Motion

When a body has both translational acceleration and angular acceleration, the total acceleration of any point on the body is given by:

where,
= translational acceleration,
= angular acceleration,
= distance from the axis of rotation.

Thus, acceleration in combined motion also has both linear and angular components.

Examples of Combined Motion

Rolling Wheel:
A wheel or cylinder rolling on a road exhibits combined motion because it translates forward and rotates about its center.
Bicycle Wheel:
The wheel of a moving bicycle rotates about its center while translating along the road.
Car Wheel:
The car wheel undergoes rolling motion, which is a typical example of combined motion.
Ball Rolling Down a Hill:
The ball moves downward due to translation and spins due to rotation.
Piston and Crank Mechanism:
The connecting rod in an engine experiences combined motion because it both rotates around the crankshaft and translates along the piston axis.

Applications in Engineering

Vehicle Dynamics:
The motion of wheels, tires, and axles in automobiles is analyzed using the principles of combined motion.
Mechanisms and Machines:
In linkages and cam-follower systems, many components exhibit combined motion during operation.
Robotics:
Robotic arms use combined translation and rotation to achieve precise movement and positioning.
Rolling Bearings:
The rolling elements in bearings perform combined motion to reduce friction between moving parts.
Design of Engines:
The piston and connecting rod motion in internal combustion engines is studied using combined motion concepts.

Advantages of Combined Motion

Provides smooth and efficient energy transfer.
Reduces friction in mechanical systems (as in rolling motion).
Enables complex mechanical operations with simple movement.
Common in designing rotating and moving parts of machinery.

Conclusion

Combined motion is the motion of a rigid body in which both translation and rotation occur simultaneously. It is the most general type of motion observed in practical mechanical systems. A rolling wheel is the best example of combined motion, as it translates and rotates together. The velocity and acceleration of different points on the body vary due to the combination of linear and angular motion. Combined motion is widely used in engineering applications such as vehicles, machines, linkages, and mechanisms, making it an essential concept in mechanical design and dynamics.