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- by Raunak Kumar almost 2 years ago

*Equilibrium*

P = constant

L = constant

An object is in static equilibrium if its linear momentum and angular momentum is equal to zero:

P = 0 kg m/s,

L = 0 kg m2/s

According to Newton’s second law of motion, the linear acceleration of a rigid body is caused by a net force acting on it, or

Here, the sum is of all external forces acting on the body, where m is its mass and a⃗ CM is the linear acceleration of its center of mass (a concept we discussed in Linear Momentum and Collisions on linear momentum and collisions). In equilibrium, the linear acceleration is zero. If we set the acceleration to zero in Equation 12.1, we obtain the following equation:

The first equilibrium condition, Equation 12.2, is the equilibrium condition for forces, which we encountered when studying applications of Newton’s laws.

(12.4)

Here I is the rotational inertia of the body in rotation about this axis and the summation is over all torques τ⃗ k of external forces in Equation 12.2. In equilibrium, the rotational acceleration is zero. By setting to zero the right-hand side of Equation 12.4, we obtain the second equilibrium condition:

The second equilibrium condition, Equation 12.5, is the equilibrium condition for torques that we encountered when we studied rotational dynamics. It is worth noting that this equation for equilibrium is generally valid for rotational equilibrium about any axis of rotation (fixed or otherwise). Again, this vector equation is equivalent to three scalar equations for the vector components of the net torque:

The second equilibrium condition means that in equilibrium, there is no net external torque to cause rotation about any axis.

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