# How can static friction do work?

By definition, the work done by a force is $W = F\cdot d$, so how can static friction do work?

Can this force move the body a distance of $75~\text{m}$?

• And the work will also be equal to the change in energy. – hft Apr 10 '15 at 4:23
• The friction is just the mechanism by which the force is transferred. For example, if the surface was frictionless, you'd expect the crate to slide right off the surface as the truck accelerated. There are other ways you could transfer the acceleration, like a rope wrapped around the box. – MonkeysUncle Apr 10 '15 at 4:33
• Static friction is being applied in an accelerating frame, that of the truck. Aside: In case (b), the bed of that flatbed truck needs to be very long. – David Hammen Apr 10 '15 at 4:33

I think you are confused about what $d$ is supposed to mean in the equation $W=F\cdot d.$ You seem to be under the impression that $d$ is the distance that the object being acted on moves relative to the object providing the force. But this is not the correct meaning of $d$ in the equation and you know it.

Imagine if the car crate were in front of the truck, and the truck were pushing the crate. Then I think you would have no problem saying that the truck is doing work on the crate even though there is no change in the relative distance between the truck and the crate.

Now the situation in your question is basically the same as this one except the force acts on the bottom of the crate instead of the side, and the force is due to friction instead of a normal force. But neither of these differences ought to change the amount of work being done.

That being said, you would have a valid point if the problem were asking for the work done in the frame of the car. In that frame, the box does not move (assuming the coefficient of static friction is sufficiently large), so that $d$ really is zero. Thus no work is done in this frame.

• So, the friction does work in moving the crate. That means, the friction is acting rightwards. But why should it act rightwards? Friction opposes the relative motion. So, was there any motion leftwards that was halted by the friction as it was acting rightwards? – user36790 Apr 11 '15 at 2:46
• Without friction, the crate would remain still as the truck begins to move. (This is a simple consequence of newton's first law.) And so without friction, the crate would move left relative to the truck. Friction opposes this motion by providing a rightward force on the crate sufficient to keep the crate and truck moving at the same velocity. – Brian Moths Apr 13 '15 at 18:26
• @user36790, see the edit to my answer – user76816 Apr 15 '15 at 9:15
• @user36790 Static friction tries to prevent the relative movement between two surfaces. In this case it makes sure that the crate keeps up with the truck and so acts in the same direction as the movement of the accelerating truck. It is kinetic friction which tries to reduce the relative movement between two surfaces. – Farcher Jul 23 '17 at 8:00

Without friction between the crate and the truck bed, the crate would remain at rest in the frame of reference of the road, as the truck accelerates away down the road.

The crate moves in the frame of reference of the road, because of the force of friction acting on it.

So the work done on the crate, in the frame of reference of the road, is the friction force times the distance the crate has traveled in the frame of reference of the road (which could be less than the distance the truck has traveled).

• can you explain this answer in mathematics ? i need more explanation,please. your answer is correct. – mech.eng Apr 10 '15 at 17:55
• @mech.eng, I'm sorry, but I'm currently hung up on the dynamics of rotations. – nir Apr 10 '15 at 19:22

By definition, the work done by a force is $W = F\cdot d$, so how can static friction do work ?

Can this force move the body a distance of $75~\text{m}$ ?

Friction does negative work on the truck, slowing it down and does not move it forward.

What does positive work on the truck, accelerates it and makes it translate $75~\text{m}$ is the engine of the truck.

The 80-Kg crate does negative work on it, because it is opposing the truck which is trying to push it forward, and acts over a distance of 75m.

Knowing that in $75~\text{m}$ of space, it reaches the velocity of $72~\text{Km/h}$, you can find the acceleration of the crate/truck. Using the weight of the crate and the coefficients of friction, you can find out the negative work done by the crate on the truck, subtracting energy and slowing it down.

Static friction locks the crate to the truck and prevents it from slipping back and off the truck's bed.

yes, this is the work done by the friction force on the truck not on the crate. I need to understand the work done by the friction force on the crate . – mech.eng

As I said, friction just locks the crate, it is an interface, like the clutch on a motor-car. The amount of positive work on the crate equals the amount of negative work $-W = +W$ on the truck. But the work on the crate is actually done by the engine of the truck.

The black arrow to the right shows the truck pulling the crate (thanks to friction) speeding it up to $v=20m/s$ and therefore giving it $W = E =16,000J$. I hope it is clear now.

It doesn't. The work is done by the active force(ex. A human trying to pull a bull.). This work is converted into frictional energy(ex. Heat generated b/w surfaces)

• Ignoring air resistance, the only force acting on the crate in the horizontal direction is a static frictional force due to the truck acting to the right which increases the momentum, velocity and kinetic energy of the crate. That static frictional force does work on the crate. Note that there is no relative movement between the crate and the truck and so no heat is generated. – Farcher Jul 23 '17 at 8:19

Static friction does not produce or consume work in most of the times. For example for a solid body that rolls without sliding the velocity of the base point $A$ is $\vec v_a = \vec v_{cm} + \vec v_{tangential} \Rightarrow v_a = v_{cm} - \omega R = \omega R - \omega R =0$ which implies that $x_a = 0$. The static friction is a force that acts on $A$ so $W_T = T x_a = 0$

But when the object slides then the friction force is constant and equal to $T = \mu N$ and is is always opposite to the velocity of the body. So then $W_T = - Ts$ where $s$ is the total space traveled by the body.

• The friction you've considered in the second part of your answer is kinetic friction - not static. The OP has asked if static friction can do work. – arya_stark Feb 9 '18 at 15:14