Force applied at centre of mass

[chandran]

If a force is applied at centre of mass the object will only translate and not rotate. Can anyone proove this mathematically?

 

[Zorodius]

Torque is the cross product of the vector from the center of mass to the point where the force is applied (r), and the force. If r is zero, or r is parallel to the force, there will be no torque, and so no rotational force.

 

[charng]

Yes, this statement can be proven mathematically using the principles of torque and rotational equilibrium. The torque, or tendency to rotate, of a force applied at a point on an object is equal to the magnitude of the force multiplied by the perpendicular distance from the point to the axis of rotation. In the case of a force applied at the center of mass, the perpendicular distance to the axis of rotation is zero, meaning that the torque is also zero. This results in the object not experiencing any rotational motion and only translating in a straight line. This can be expressed mathematically as τ = F * r * sin(θ) = 0, where τ is the torque, F is the force, r is the distance from the center of mass to the axis of rotation, and θ is the angle between the force and the line connecting the center of mass to the axis of rotation. Therefore, a force applied at the center of mass will not cause the object to rotate.