Mechanics 8.1 - Force and Newton's first law

Mechanics 8.1 - Force and Newton's first law

Learn to name the forces on one body, add them with their directions and use Newton's first law to decide whether its velocity changes. You will build force diagrams and use zero-resultant equations for objects at rest and objects already moving.

Forces as vectors

Can a trolley keep moving when the resultant force on it is zero? Imagine a trolley already travelling right at 4 m s−14\,\mathrm{m\,s^{-1}}. A pull of 12 N12\,\mathrm N acts right and a resistance of 12 N12\,\mathrm N acts left. The two forces cancel. Does the trolley stop, or carry on? We will settle that question by distinguishing a force from its resultant.

You already know

A vector has magnitude and direction. Opposite directions have opposite signs on a chosen axis; for example, 20 N20\,\mathrm N right and 20 N20\,\mathrm N left represent different vectors.

Choose the body before drawing arrows. A force exerted on the trolley belongs in its diagram; a force the trolley exerts on the floor belongs in the floor's diagram.

Force and resultant force

A force is a push or pull on a body, measured in newtons, N\mathrm N. It is a vector. The resultant force is the vector sum of all forces acting on the chosen body. For example, opposite 20 N20\,\mathrm N forces have zero resultant; two 20 N20\,\mathrm N forces in the same direction have resultant 40 N40\,\mathrm N.

For forces along one line, signs do the vector work. Write which direction is positive before subtracting.

Worked example 1: opposing pulls

Question. The only horizontal forces on a trolley are 20 N20\,\mathrm N right and 12 N12\,\mathrm N left. Find the horizontal resultant.

Plan. Take right as positive and add the signed forces.

F=20−12leftward force is negative=8 Nadd the components\begin{aligned} F&=20-12 &&\text{leftward force is negative}\\ &=8\,\mathrm N &&\text{add the components} \end{aligned}

Why subtract? The forces act along the same line in opposite directions.

Answer. The resultant is 8 N8\,\mathrm N to the right.

Check. The larger force points right, and 8+12=208+12=20.

Now reverse which pull is larger. That changes the sign of the answer, not the addition rule.

Worked example 2: a negative component

Question. The only horizontal forces on a trolley are 12 N12\,\mathrm N right and 20 N20\,\mathrm N left. Find the horizontal resultant.

Plan. Keep right positive so that the sign identifies the direction.

F=12−20use the same positive direction=−8 Nadd the signed forces\begin{aligned} F&=12-20 &&\text{use the same positive direction}\\ &=-8\,\mathrm N &&\text{add the signed forces} \end{aligned}

Why keep the minus sign? It tells us that the resultant points opposite to right.

Answer. The resultant is 8 N8\,\mathrm N to the left.

Check. Taking left positive instead gives 20−12=8 N20-12=8\,\mathrm N, the same physical result.

A resultant is a signed component or a vector; its magnitude is non-negative.

Watch out

The line F=12+20=32 NF=12+20=32\,\mathrm N ignores direction. For the opposing forces in example 2, the correct rightward component is 12−20=−8 N12-20=-8\,\mathrm N.

Try another opposing pair, keeping the direction in your conclusion.

Common force names

The name of a force helps you draw its direction before you know its magnitude. Picture a particle pulled right along a rough horizontal table by a taut horizontal string: the string pulls right, the resistance acts left, the table pushes up and the weight acts down.

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Diagram

Each arrow describes an interaction with something outside the chosen body.

Common forces

A normal reaction is a contact force perpendicular to the surface. Tension is a pulling force along a taut string. Thrust or compression is a push along a rod or contact line. Resistance, such as drag or friction, opposes motion or the tendency for relative motion. Weight is the gravitational force, vertically downwards near Earth's surface.

For a block on a slope, the reaction is tilted because it is normal to the slope; the weight is still vertical. A slack string exerts no tension, and a string cannot push. Resistance is not automatically leftwards: reverse the motion and its direction reverses. If the weight is supplied in newtons, use that value directly; a mass in kilograms is a different quantity.

Key idea

Draw only forces acting on your chosen body, with directions determined by the interaction: normal means perpendicular, and tension means a pull along the string.

Use the physical description to distinguish a force direction from a motion direction.

Newton's first law

A trolley can have several forces acting on it while its velocity stays unchanged. What matters is their sum. In our opening case, 12−12=012-12=0, so there is no change to the trolley's 4 m s−14\,\mathrm{m\,s^{-1}} rightward velocity.

[DIAGRAM: asset_slug: p3_8_1_understand_the_concept_of_force_and_understand_and_use_newton_s_first_law__diagram_02; asset_name: Mechanics 8.1 - Force and Newton's first law - diagram 02; recommended_method: drawn_math; description: Draw a two-panel NovaLearn style diagram. Left panel: a block at rest with equal 12 N12\,\mathrm{N} forces to the left and right, labelled resultant force = 00 and stays at rest. Right panel: a block already moving right at constant 4 m s−14\ \mathrm{m}\,\mathrm{s}^{-1} with equal 12 N12\,\mathrm{N} forces to the left and right, labelled resultant force = 00 and continues at the same velocity.]
Diagram

The two cases have the same force balance but different initial velocities.

Newton's first law

For a body in the usual fixed-frame mechanics model,

∑F=0⟹v is constant.\sum\mathbf F=\mathbf0\quad\Longrightarrow\quad\mathbf v\text{ is constant}.

The body remains at rest or continues at constant speed in a straight line. Not in the formula booklet: learn it.

Constant velocity includes direction. Constant speed around a bend is not constant velocity. A particle in equilibrium has zero resultant force; it need not be stationary.

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