Spherical Motion of a Rigid Body

Spherical motion of a rigid body is three-dimensional motion in which one point of the body remains fixed. Every other point moves on a spherical surface centered at that fixed point.

Geometry of spherical motion

Let point $O$ be fixed. The distance $OP$ to any body point $P$ is constant, so the trajectory of $P$ lies on a sphere of radius $OP$. Unlike rotation about a fixed axis, the direction of the instantaneous rotation axis generally changes with time.

Instantaneous angular velocity

For a rigid body with a fixed point $O$, the velocity of any point $P$ at an instant is determined by the instantaneous angular velocity:

$$\vec v_P=\vec\omega\times\vec r_{P/O}.$$

Points on the instantaneous axis of rotation have zero velocity at that instant. For other points, velocity is perpendicular to the plane formed by $\vec\omega$ and $\vec r_{P/O}$.

The vector $\vec\omega$ describes the instantaneous rotation of the body. The line through fixed point $O$ in the direction of $\vec\omega$ is the instantaneous axis of rotation.

Point velocities

The speed of point $P$ is $v_P=\omega r_\perp$, where $r_\perp$ is the perpendicular distance from the point to the instantaneous axis. Points on the instantaneous axis therefore have zero velocity at that instant.

Changing instantaneous axis

The instantaneous axis is not generally one material line that remains fixed throughout the motion. Its position changes both within the body and in fixed space. This distinguishes general spherical motion from simple fixed-axis rotation.

Angular acceleration

Angular acceleration is $\vec\varepsilon=d\vec\omega/dt$ in the fixed reference frame. Because both magnitude and direction of $\vec\omega$ may change, $\vec\varepsilon$ and $\vec\omega$ are not generally parallel.

Example

If at an instant $\omega=6$ rad/s and a point is at a perpendicular distance of 0.15 m from the instantaneous axis, its speed is $v=6\cdot0.15=0.9$ m/s.

Common mistakes

• assuming the instantaneous axis remains fixed throughout spherical motion;
• using the full distance $OP$ instead of perpendicular distance to the instantaneous axis in $v=\omega r_\perp$;
• assuming $\vec\varepsilon$ is always parallel to $\vec\omega$;
• confusing spherical rigid-body motion with the motion of a single particle on a sphere.