Why does it take a bigger push to start a crate moving than to keep it moving?

Two frictions, not one

While you push a crate that is not moving, static friction balances your push exactly. Push more gently and it is gentler; push harder and it is harder. It adapts.

But it has a limit. Beyond its maximum it can no longer balance, the crate breaks free, and from that moment kinetic friction acts — constant, and usually smaller than the static maximum.

Both are found the same way: the coefficient of friction times the normal force. The static coefficient is larger than the kinetic one, and that is the entire explanation for the lurch you feel as the crate starts to move.

What the normal force really is

On a horizontal surface with no vertical pushes the normal force equals the weight, which makes it easy to mistake that for its definition. It is not.

Press down on the crate and the normal force grows, and the friction with it. Pull at an upward angle and the normal force shrinks and the friction shrinks — which is why a suitcase is easier to pull than to push.

On a slope the normal force is the component of weight perpendicular to the surface, that is the weight times the cosine of the angle. The steeper the slope, the smaller the normal force, the weaker the friction, and the larger the component pulling downhill — two effects working the same way.

Two masses over a pulley

Two bodies joined by a string move together, so they share one acceleration in magnitude. The string carries the same tension at both ends when it is light and the pulley is smooth.

The method is to write the second law for each body separately, then solve two equations for two unknowns — the acceleration and the tension. A useful shortcut for the system as a whole: the acceleration is the driving force minus the opposing force, divided by the total mass.

The tension is not equal to the weight of the hanging body unless the system is not accelerating. That is the single most repeated mistake in these questions.

Worked examples

1. A 20 kg crate on a horizontal floor, kinetic coefficient 0.3. What is the friction while it moves?

   1. The normal force equals the weight: 20 times 10, so 200 N
   2. Friction is the coefficient times the normal force
   3. 0.3 times 200

   Answer: 60 N, opposite to the motion

2. A 3 kg mass on a smooth table is joined over a pulley to a 2 kg hanging mass. What is the acceleration?

   1. The driving force is the hanging weight: 2 times 10 = 20 N
   2. There is no friction, so nothing opposes it
   3. The total mass is 3 plus 2, which is 5
   4. The acceleration is 20 divided by 5

   Answer: 4 m/s², and both masses move together

3. In the same system, what is the tension in the string?

   1. Write the second law for the body on the table
   2. The only force along the motion is the tension
   3. Tension equals 3 times 4

   Answer: 12 N, not 20 — the system is accelerating

Common mistakes

Assuming the tension equals the hanging body's weight

That holds only with no acceleration. In an accelerating system the tension is always less than the weight, or the hanging body would not descend.

Always substituting weight for the normal force

They are equal only on a horizontal surface with no other vertical forces. Pulling at an angle or tilting the surface changes it, and the friction with it.

Using the static coefficient once the body is moving

At the moment of motion kinetic friction takes over, and its coefficient is lower. Using the static one gives too much friction and too little acceleration.

More in Newton's Laws

• Why does every mechanics problem start with a free-body diagram?
• If there is no net force, why does a body keep moving?
• Does the net force set a body's speed, or its acceleration?
• Why do a heavy body and a light one fall at the same rate?
• If every force has an equal and opposite partner, how does anything move?