Planar Kinematics of a Rigid Body

Planar rigid-body motion describes the movement of a rigid body while its particles stay at a constant distance from a plane. This idea helps explain how objects move in physics and mechanical engineering. It also gives a clear way to study motion in space.

One type of planar motion is rectilinear translation. In this motion, the body moves in a straight line. Any two points on the body follow parallel paths, like a subway train moving along its track or soldiers marching in step.

Another...

Angular motion around a fixed axis describes how a rigid body spins around one line. As it rotates, each point in the body follows a circular path around that line or point. The motion is described using angular position, written as theta (θ), which is measured from a fixed reference line.

When angular position changes, the change is called angular displacement, written as dθ. Angular displacement can be measured in degrees, radians, or revolutions. One revolution equals 2π radians. This value...

Rotation kinematics describes the motion of a rigid body when it turns with constant angular acceleration. It follows the same basic ideas used in linear kinematics, but for circular motion instead of straight-line motion.

For a point A on the rotating body, translational velocity comes from the time derivative of the displacement equation. This velocity is always tangent to the circular path of point A. It can also be written as the vector product of angular velocity and the position vector.

Drone flight shows general plane motion, which combines translation and rotation. A drone lifts off when its spinning propellers create an upward force that balances gravity. This first movement is straight-line translational motion. During this stage, every point on the drone moves the same distance in the same direction.

Once the drone is airborne, its motion becomes more complex. A point on the drone, such as point P, keeps moving while the drone flies. Point P follows a curved path instead...

A slider-crank mechanism converts rotational motion from a crank into linear motion of a slider, or the reverse. It has three main parts: the crank, the connecting rod, and the slider. When an external force acts on the system, the crank begins to rotate and drives the motion of the connecting rod.

The connecting rod moves with general plane motion. In this motion, point A on the rod and point B on the crank both have translational motion. Point B also has rotational motion relative to point...

General plane motion can be seen in a rolling wheel. The wheel rotates and translates at the same time. To study this motion, the velocity of one point can be written as the vector sum of the velocity of another point and the relative velocity between them.

A useful simplification is to choose a point that has zero velocity at a given instant. This point is called the instantaneous center of zero velocity, or IC. The IC lies on an axis perpendicular to the plane of motion. Where that axis...

A slider-crank mechanism turns rotational motion from the crank into linear motion of the slider, and it can also work the other way around. It has three main parts: the crank, the connecting rod, and the slider. Because the angle between the crank and the connecting rod keeps changing, the motion is an example of general plane motion.

To study the motion, a stationary reference system is placed at point O, and a translating frame is added at point A. The segment AB is then analyzed, with...

A rotating frame of reference helps describe the motion of point B on a moving member AB. The member has linear motion, and point B also rotates around point A. To track that motion, position vectors for A and B are first written in a fixed reference frame.

A second frame, x'y', is then used to show the position of point B relative to point A. This frame both translates and rotates with the system. That makes it useful for analyzing the relative position more clearly.

The velocity of point B...

Acceleration of point B is found by analyzing a member AB that moves linearly and also rotates about point A. The motion is described with position vectors for points A and B in a fixed reference frame. This setup helps separate the motion of the point from the motion of the rotating frame.

The absolute velocity of point B is written as the velocity of point A plus the relative velocity of point B in the rotating frame. It also includes the effect of angular velocity in that rotating frame. To...

Crane boom motion on rotating axes combines rotation and extension in one kinematics problem. A telescopic boom rotates while it also extends outward. Relative motion analysis helps find the velocity and acceleration of the moving end of the boom.

In the setup, the boom rotates with an angular velocity of 0.04 rad/s and an angular acceleration of 0.02 rad/s². It also extends linearly at a constant speed of 5 m/s. The extension is measured at point D with respect to the fixed point C at the...

Flywheel motion about a fixed axis depends on how its mass is spread out and how it accelerates. In this case, the flywheel has an uneven mass distribution and rotates steadily around a fixed axis. As it spins, its center of mass follows a circular path.

The acceleration of the center of mass has two parts. The tangential component points along the path and depends on the direction of the flywheel’s angular acceleration. It helps move the flywheel along its circular path. The normal component...

Rigid body motion in a general plane is driven by external forces and couple moments. The body moves and turns at the same time. Newton’s second law describes the translational motion of the center of mass along each axis.

The center of mass also rotates because of the applied couple moments. That rotation is described by angular acceleration and moment of inertia. These two quantities connect the turning effect to how hard the body is to rotate.

The same motion equations can be written for...

A lawn roller’s angular acceleration can be found by using rolling motion and friction. The roller in this problem has a mass of 100 kg, a radius of 0.2 meters, and a radius of gyration of 0.15 meters. A force of 200 N is applied at an angle of 60 degrees from the horizontal.

The setup also includes friction between the roller and the ground. The static friction coefficient is 0.15, and the kinetic friction coefficient is 0.1. The motion is treated as rolling without slipping, so the point of...