Zaid Berjis

PhysicsLesson 2 of 18About 40 minutes

Inertia, net force and equilibrium

If you stop pedaling a bicycle on a flat road, you slowly coast to a stop. It is natural to conclude that moving things need a force to keep them moving. Newton's first law says the opposite, and this lesson shows why it is right.

By the end of this lesson you can

  • explain inertia and state Newton's first law in your own words;
  • explain why mass is a measure of inertia;
  • find the net force on an object from its free-body diagram;
  • decide whether an object is in equilibrium, and find the force that would put it in equilibrium.

Inertia

Think first

You give a hockey puck one quick push on a smooth ice rink. Once your stick leaves it, is anything pushing the puck forward? Why does it keep going for so long?

Compare your answer

Nothing pushes it forward once the stick leaves. It keeps moving because nothing much is stopping it: ice gives very little friction. Moving objects do not need a forward force to keep moving. They need a force to change their motion.

Definition

Inertia is an object's resistance to any change in its motion. A resting object stays put; a moving object keeps the same speed and direction.

The bicycle slows down because forces act on it: friction in the wheels and the push of the air. Take those forces away and it would roll on forever at the same speed. You can see this by sliding the same block, with the same push, over smoother and smoother floors.

One push, three floors

Interactive
\(\vec F_{\text{push}}\)

Scroll sideways to see the whole demo, or turn your phone sideways.

Try this

  1. Push the block on the thick carpet, then on the waxed floor. Where does it slide farther?
  2. Now choose the perfectly smooth floor. What happens to its speed after the push ends?

Newton's first law

Galileo was the first to argue, in the 1600s, that an object keeps moving unless something stops it. Newton built on that idea and made it the first of his three laws of motion.

Newton's first law (the law of inertia)

An object stays at rest, or keeps moving in a straight line at constant speed, unless a net external force acts on it.

Two words in that statement matter a lot:

  • Net means the total of all the forces, taking directions into account. Several forces can act on an object and still add up to zero.
  • External means the force comes from outside the object. You cannot lift yourself by pulling on your own belt.

Common mistake

"No net force means the object is at rest." Not quite. No net force means constant velocity. That can be zero velocity (at rest), but it can just as well be 25 m/s in a straight line.

A ball rides on a cart that hits a wall

Interactive

Scroll sideways to see the whole demo, or turn your phone sideways.

Try this

  1. Predict first: the wall stops the cart suddenly. Where does the ball go?
  2. Press Run. Then switch the ball from light to heavy and run it again.
  3. Explain what you saw using the word inertia. What force stopped the cart? Did anything stop the ball?

Quick check

A delivery truck rolls along a straight highway at a steady 25 m/s, engine on. How large is the net force acting on it?

  1. Forward, because the engine is running
  2. Backward, because air resistance acts on it
  3. Zero
  4. It depends on the mass of the car
Show solution

Constant speed in a straight line means constant velocity, so the acceleration is zero and the net force is zero. The forward push of the road on the tires is exactly balanced by air resistance and friction.

Mass is a measure of inertia

Give a basketball and a bowling ball the same shove. The basketball speeds up much more. The bowling ball "resists" the change in its motion more strongly: it has more inertia.

The quantity that measures inertia is mass. The more mass an object has, the smaller the change in its velocity for the same push. That is why a loaded shopping cart is harder to start and harder to stop than an empty one.

The same push on two different masses

Interactive
\(\vec F\)
\(\vec F\)

Scroll sideways to see the whole demo, or turn your phone sideways.

Try this

Press Push both equally and compare the speeds the two balls end up with. Which ball has more inertia, and how does the demo show it?

Net force

A car cruising at constant speed has at least four forces on it: gravity down, the road up, a forward push from the road on the drive wheels, and air resistance backward. Its velocity does not change because those forces cancel out.

Definition

The net force on an object is what you get when you add up every external force on it as vectors. It is the single force that would have the same effect as all of them together. We write it \(\vec F_{\text{net}}\) or \(\sum \vec F\).

Add up four forces

Interactive
\(\vec F_1\)
\(\vec F_2\)
\(\vec F_3\)
\(\vec F_4\)
\(\vec F_{\text{net}}\)
Right \(F_1\)Left \(F_2\)
Up \(F_3\)Down \(F_4\)

Scroll sideways to see the whole demo, or turn your phone sideways.

Try this

  1. Press Car at constant velocity. What is the net force?
  2. Make the right-hand force 2 N larger than the left. Which way does the red net force arrow point, and how long is it?
  3. Find a setting where the net force points up and to the left.

Finding the net force

  1. Draw the free-body diagram and choose \(x\) and \(y\) axes. Pick them so that as many forces as possible lie along an axis.
  2. Split every slanted force into components: \(F_x = F\cos\theta\) and \(F_y = F\sin\theta\), with \(\theta\) measured from the \(x\)-axis.
  3. Add the \(x\) components to get \(F_{\text{net},x}\) and the \(y\) components to get \(F_{\text{net},y}\). Use signs for direction.
  4. Combine: \(F_{\text{net}} = \sqrt{F_{\text{net},x}^2 + F_{\text{net},y}^2}\), at an angle \(\theta = \tan^{-1}\left(\dfrac{F_{\text{net},y}}{F_{\text{net},x}}\right)\).

Worked example 1

A small boat's motor pushes it with 600 N toward the east. At the same time the river current pushes it with 250 N toward the south. How large is the net force on the boat, and which way does it point?

  1. Choose axes. Let \(+x\) be east and \(+y\) be north. Then the motor force is \(+600\ \text{N}\) along \(x\) and the current is \(-250\ \text{N}\) along \(y\).
  2. Add the components. \(F_{\text{net},x} = 600\ \text{N}\) and \(F_{\text{net},y} = -250\ \text{N}\). Each axis has only one force, so nothing else to add.
  3. Find the size. $$F_{\text{net}} = \sqrt{(600\ \text{N})^2 + (250\ \text{N})^2} = 650\ \text{N}$$
  4. Find the direction. $$\theta = \tan^{-1}\left(\frac{250}{600}\right) = 22.6^\circ$$ The net force points south of east.

    \(F_{\text{net}} = 650\ \text{N}\), at 22.6° south of east

Worked example 2

A crate rests on a ramp tilted at 25.0°. Three forces act on it: gravity (40.0 N, straight down), friction from the ramp (9.0 N, up the slope), and the ramp's push perpendicular to its surface (36.3 N). Find the net force on the crate.

  1. Tilt the axes. Let \(+x\) point down the slope and \(+y\) point straight out of the ramp. Now friction and the ramp's push each lie along an axis; only gravity is slanted.
  2. Split gravity into components. Relative to the tilted axes, gravity makes an angle of 25.0° with the \(-y\) direction: $$F_{g,x} = (40.0\ \text{N})\sin 25.0^\circ = 16.9\ \text{N} \quad\text{(down the slope)}$$ $$F_{g,y} = -(40.0\ \text{N})\cos 25.0^\circ = -36.3\ \text{N}\quad\text{(into the ramp)}$$
  3. Add along each axis. $$F_{\text{net},x} = 16.9\ \text{N} - 9.0\ \text{N} = 7.9\ \text{N}$$ $$F_{\text{net},y} = 36.3\ \text{N} - 36.3\ \text{N} = 0$$
  4. State the result. The perpendicular forces cancel. What is left points down the slope.

    \(F_{\text{net}} = 7.9\ \text{N}\) down the slope, so the crate speeds up as it slides.

Equilibrium

Definition

An object is in equilibrium when the net force on it is zero: \(\sum \vec F = 0\). In components, \(\sum F_x = 0\) and \(\sum F_y = 0\).

A book resting on a shelf is in equilibrium. So is a skydiver falling at a steady speed, and so is the car cruising on the highway. If you find that the net force on an object is not zero, you can always bring it into equilibrium by adding one more force: equal in size to the net force and pointing the opposite way.

Worked example 3

Two children pull a sled across flat, slippery ice with ropes. One pulls with 80 N toward the east, the other with 60 N toward the north. A parent wants to hold the sled still with one rope. What force must the parent apply?

  1. Find the net force from the children. The pulls are perpendicular: $$F_{\text{net}} = \sqrt{(80\ \text{N})^2 + (60\ \text{N})^2} = 100\ \text{N}$$ at \(\tan^{-1}(60/80) = 36.9^\circ\) north of east.
  2. Apply the equilibrium condition. For the sled to stay still, the parent's force plus the net force must be zero, so the parent's force must be the exact opposite of the net force.
  3. Reverse the direction. The opposite of "north of east" is "south of west".

    100 N, at 36.9° south of west

Check yourself

Question 1

A bus brakes suddenly and the standing passengers lurch forward. What explains this?

  1. The brakes push the passengers forward.
  2. The passengers keep moving forward while the bus slows beneath them.
  3. Air inside the bus pushes the passengers forward.
  4. Gravity pulls the passengers toward the front of the bus.
Show solution

No forward force acts on the passengers. Their inertia keeps them moving at the bus's old speed, while friction from the floor on their feet is not enough to slow them as quickly as the bus slows.

Question 2

Wind pushes a falling leaf with 3.0 N to the east while gravity pulls it with 4.0 N downward. What is the size of the net force on the leaf? (Ignore air resistance.)

Show solution

The forces are perpendicular: \(F_{\text{net}} = \sqrt{3.0^2 + 4.0^2} = 5.0\ \text{N}\), at \(\tan^{-1}(3.0/4.0) = 37^\circ\) east of straight down.

Question 3

In a tug-of-war, one team pulls the rope with 520 N to the left and the other with 500 N to the right. How large a force, added to the weaker side, would put the rope in equilibrium?

Show solution

\(\sum F = 520\ \text{N} - 500\ \text{N} = 20\ \text{N}\) to the left, so the rope is not in equilibrium. Adding 20 N to the right makes the net force zero.

Question 4

A student pulls a suitcase with a strap at 40.0° above the horizontal, using a force of 61.4 N. What is the horizontal component of the pull?

Show solution

\(F_x = F\cos\theta = (61.4\ \text{N})\cos 40.0^\circ = 47.0\ \text{N}\). The vertical component is \((61.4\ \text{N})\sin 40.0^\circ = 39.5\ \text{N}\).

Question 5

A tossed coin is momentarily at rest at the highest point of its flight. Is it in equilibrium at that moment?

Show solution

No. Its velocity is zero for an instant, but gravity still acts on it, so the net force is not zero and its velocity keeps changing. Equilibrium is about the net force, not about the velocity at one moment.

Summary

  • Inertia is the tendency of an object to keep its state of motion. Mass measures inertia.
  • Newton's first law: unless a net external force acts, an object's velocity does not change, whether that velocity is zero or not.
  • Add all the forces on an object as vectors, using components, to get the net force.
  • An object is in equilibrium when \(\sum \vec F = 0\). Adding one force that exactly cancels the net force brings it into equilibrium.

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