6.2

First Law: Particles in Two-dimensional Equilibrium

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An object is in two-dimensional equilibrium when forces acting on it in two directions are balanced.

Suppose a mass is suspended by two strings. The forces acting on the mass are its own weight acting downward, tension along the first string, and tension along the second string.

As the mass is in equilibrium, we consider magnitudes of horizontal and vertical components separately and add them to zero.

Now, consider two masses m₁ and m₂ at rest, connected by a light string going over a smooth, light pulley on an inclined surface.

Here, we consider both masses separately. The forces acting on m₁ are its weight, tension along the string, and the normal force due to the inclined surface. These force magnitudes should add up to zero and keep m₁ at rest.

The forces acting on m₂ are its weight acting downward and the string's tension acting upward. As m₂ is at rest, the net magnitude of force acting on it should be zero.

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$$\rightleftharpoonup{xx}$$ $$\longleftharp{xx}$$, $$\longrightharp{xx}$$,

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration; but it is important to remember that these conditions are relative. For instance, an object may be at rest when viewed from one frame of reference, but that same object would appear to be in motion when viewed by someone moving at a constant velocity.

Newton's first law tells us about the equilibrium of a system, which is the state in which the forces on the system are balanced. To create equilibrium, we require a balancing force that will produce zero net force. This force must be equal in magnitude but opposite in the direction to the resultant force.

Newton's laws help solve mechanical problems. If a problem is two-dimensional, it must be broken down into a pair of one-dimensional problems. We can do this by projecting the force vectors onto a convenient set of axes. It is almost always convenient to make one axis parallel to the direction of motion if this is known. Generally, just write Newton's second law in components along the different directions. If, for example, the system is accelerating horizontally, then set the vertical component of acceleration to zero. We need this information to determine the unknown forces acting on a system.

The following steps can be applied to find a solution.

Identify the physical principles involved by listing the known quantities and the quantities to be calculated.
Sketch the situation, using arrows to represent all forces.
Determine the system of interest. The result is a free-body diagram that is essential to solving the problem.
Apply Newton's second law to solve the problem. If necessary, apply appropriate kinematic equations from the chapter on motion along a straight line.
Check the solution to see whether the solution is reasonable.

This text is adapted from Openstax, University Physics Volume 1, Section 6.1: Solving Problems with Newton's Laws.

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