Gravitational torque is the turning effect produced by the force of gravity acting on a body whose center of mass is not directly aligned with its axis or point of rotation. Torque is a vector quantity that measures the tendency of a force to cause rotational motion. In the case of gravitational torque, the force involved is the weight of the object, which acts vertically downward through the object's center of mass. When the line of action of this weight does not pass through the pivot or axis of rotation, a moment arm exists, and gravity generates a torque that tends to rotate the object.
Gravitational Torque Formula |
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| \( \tau_g \;=\; m \cdot g \cdot r \cdot sin(\theta) \) |
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| Symbol | English | Metric |
| \( \tau_g \) (Greek symbol tau) = Gravitational Torque | \(lbf-ft\) | \(N-m\) |
| \( m \) = Object Mass | \(lbm\) | \(kg\) |
| \( g \) = Gravitational Acceleration (See Physics Constant) | \(ft \;/\; sec^2\) | \(m \;/\; s^2\) |
| \( r \) = Radius (Distance from the Axis of Rotation to the Center of Mass of the Object) | \(ft\) | \(m\) |
| \( \theta \) = Angle (Angle between the Line Connecting the Center of Mass to the Axis of Rotation and the Direction of the Gravitational Force) | \(deg\) | \(rad\) |
The magnitude of gravitational torque is equal to the product of the gravitational force (weight) and the perpendicular distance from the axis of rotation to the line of action of that force. Mathematically, torque is expressed as the cross product of the position vector and the gravitational force. The direction of the torque depends on the orientation of the object and determines whether the resulting rotation is clockwise or counterclockwise about the pivot.
Gravitational torque is a concept in statics and dynamics. It explains why a suspended pendulum swings toward its lowest position, why a tilted object may tip over, and why structures and mechanical systems must be designed to maintain stability. When the center of mass lies directly below or above the pivot so that the line of action of the weight passes through the axis of rotation, the gravitational torque is zero because there is no moment arm. Under such conditions, gravity does not create a tendency for the object to rotate about that axis.
Gravitational Torque Examples
- A simple pendulum experiences gravitational torque when displaced from its vertical equilibrium position. The weight of the pendulum bob acts through its center of mass, creating a torque about the pivot that causes the pendulum to swing back toward equilibrium.
- A seesaw experiences gravitational torque when the weights on its two sides are unequal. The heavier side creates a greater torque about the pivot, causing the seesaw to rotate.
- A crane boom raised at an angle experiences gravitational torque because the boom's weight acts at its center of mass. This torque must be resisted by the crane's structure and lifting mechanisms.
- A robotic arm or mechanical linkage experiences gravitational torque when positioned away from its vertical orientation. The arm's weight creates a turning moment about its joints that actuators must overcome to maintain position.
- A sign or beam supported at one end as a cantilever experiences gravitational torque due to its own weight acting at its center of mass. The support structure must resist this torque to prevent rotation.
- A satellite or spacecraft in a gravitational field can experience gravitational gradient torque when different parts of the vehicle are at slightly different distances from the attracting body. This torque tends to align the spacecraft with the local gravitational field.
