Equilibrium and Newton's First Law
For a particle, zero resultant force means constant velocity, including rest. Balance force components in two perpendicular directions to find unknown forces.
What a learner can do afterwards
- Show three given forces are in equilibrium by resolving in two directions
- Find the tension in each of two strings holding a hanging mass
- Name every force on a book resting on a table and say what exerts it
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You have resolved forces into perpendicular components. Now use them to test equilibrium. Work in a frame that is not accelerating or rotating. For a particle, equilibrium means zero resultant external force. Its velocity stays constant: it remains at rest or moves at constant speed in a straight line. Take east and north as positive. Add signed horizontal components and signed vertical components separately. Both totals must be zero.
A book resting on a level table, with no other applied forces, has weight downward from Earth and a normal force upward from the table. Normal means perpendicular to the surface. If a hand also presses down, the normal force balances the weight plus that push. Weight and normal both act on the book. They are not an action-reaction pair: such a pair acts on two different bodies. The book's push on the table acts on the table.
Weight is mg downward, where m is mass and g is gravitational field strength. A taut string pulls along its length toward its support. For a mass at rest on one vertical string, tension T = mg. For two vertical strings, T_L + T_R = mg. Each tension equals mg/2 only if equal sharing is stated. Without that condition, force balance determines their sum but not each tension.
If the existing horizontal and vertical totals are R_x and R_y, a balancing force has components -R_x and -R_y. Its magnitude is sqrt(R_x² + R_y²), because the components are perpendicular. These are two components of one extra force. Check both signed totals after adding it. To test three given forces, add all three horizontal components and all three vertical components.
For taut strings sloping up left and up right, let a and b be their angles above the horizontal. Let T_L and T_R be their tensions and W the weight. Horizontal balance gives T_L cos(a) = T_R cos(b). Vertical balance gives T_L sin(a) + T_R sin(b) = W. Use the horizontal equation to express one tension in terms of the other, then substitute into the vertical equation. Equal angles give equal tensions.
An 8 kg particle rests on two strings, each at 30° above the horizontal on opposite sides. Use g = 10 N/kg, so W = 80 N. Horizontal balance gives T_L = T_R = T. Since sin30° = 1/2, vertical balance gives T/2 + T/2 = 80. Each tension is 80 N, but each upward component is only 40 N.
Keep W = 80 N, but use a left angle of 30° and a right angle of 60°. Use sin30° = cos60° = 1/2 and cos30° = sin60° = √3/2. Horizontally, T_L√3/2 = T_R/2, so T_R = √3 T_L. Substitute vertically: T_L/2 + 3T_L/2 = 80. Thus T_L = 40 N and T_R = 40√3 N. Their upward components are 20 N and 60 N; their horizontal components cancel.
Being motionless for one instant does not prove equilibrium. At the highest point of a vertical throw, a ball's velocity is briefly zero. Ignoring air resistance, its weight still acts downward with no balancing force. Its velocity therefore changes. Equilibrium requires zero resultant force, not just a moment with zero velocity.
For a particle, zero resultant force means constant velocity, including rest. List forces on the chosen body, then balance signed components in both directions.
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What zero resultant force tells you
Use a frame of reference that is not accelerating or rotating. When a particle has zero resultant external force, its velocity stays unchanged. Its starting velocity determines what happens next: it either remains at rest or travels at the same speed in the same direction.
Friction explains why a sliding object often slows down. The object does not need an unbalanced forward force to keep moving. It needs zero horizontal resultant to keep its horizontal velocity unchanged.
Inertia describes resistance to changes in velocity. Mass measures inertia and is measured in kilograms. Weight is different: it is a gravitational force measured in newtons. Changing the gravitational field changes an object's weight without changing its mass.
Force balance and turning
A particle model leaves an object's size and rotation out of the calculation. For that model, checking equilibrium means checking that the forces add to zero.
An extended body can turn as well as move from place to place. Two forces can cancel as vectors yet still produce a turning effect. Full equilibrium therefore needs a second check: the total turning effect, called torque, must also be zero.
Zero resultant force means no change in translational velocity. Zero total torque is the rotational equilibrium condition. These conditions do not require the object to be motionless. Static equilibrium describes the special case in which it is at rest and stays at rest.
The questions in this topic use force balance. They do not require torque calculations.
Which body feels the force?
Start a force list by choosing one body. For a book resting on a level table, the list includes Earth's downward gravitational pull and the table's upward contact force. The contact force is normal to the surface, meaning perpendicular to it.
If these are the only forces, the normal force equals the book's weight. A hand pressing down adds another downward force, so the table must push upward with a larger normal force to maintain rest.
A string has a different direction rule. When taut, it pulls along its length toward its support. For a sloping string, only the upward component of its tension supports the weight.
The book also pushes on the table, but that force belongs to the table's force list. The table's push on the book and the book's push on the table form an action-reaction pair. A pair of this kind acts on two different bodies. The book's weight and normal force both act on the book, so they are not such a pair.
Check the answer in both directions
Choose two perpendicular directions and define the positive signs. With east and north positive, westward and southward components are negative.
For equilibrium, every force on the chosen body contributes to the component sums. Add all horizontal components, then all vertical components. Both answers must be zero.
For a hanging particle, include the downward weight along with the tensions. A tilted tension T at an angle a above the horizontal has an upward component T sin(a). Its horizontal component has magnitude T cos(a), with its sign set by the direction of the pull.
After solving for unknown tensions, put the results back into both equations. This checks more than the arithmetic: it also checks whether you used the correct component directions. A pair of values that balances only one direction is not an equilibrium solution.
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