A pickup truck is carrying a toolbox, but the rear gate of the truck is missing. The toolbox will slide out if it is set moving. The coefficients of kinetic friction and static friction between the box and the level bed of the truck are 0.355 and 0.650, respectively. Starting from rest, what is the shortest time this truck could accelerate uniformly to 30.0 m/s without causing the box to slide? Draw a free-body diagram of the toolbox.

Video transcript

Welcome back everybody. We have a crate that is sitting on the plates of a forklift. Now, it is going to be moving in this direction, the forklift, that is. And we want to figure out how fast that forklift can move so that the crate doesn't slip off. Right? So, first, let's observe this crate here in in isolation, I'm actually gonna draw the forces that are acting on it right now, in order for it to not slip off. That means that there is a force acting in the direction of motion and it is a force due to friction. Now, since it is not moving on top of those plates or that is our goal, this is going to be a force due to static friction right now, of course, with any object to we have a normal force and then a force due to gravity. Now we are told a couple of other different things, right? We are told that uh forklift is going to accelerate from zero m per second to 15 m per second. And just one more time, just to reiterate, we want to find how fast or the lowest time it can do that without this guy falling off in order to find this time, we still need one more variable to use our Kinnah Matic formulas. So, I'm actually gonna go for acceleration here, and you'll see why in just one second. Well, first, let's start looking at our forces acting on this. I'm actually gonna solve for this guy right here are force due to static friction. The force due to static friction is equal to r coefficient of static friction times our normal force. But what is our normal force using Newton's Newton's laws here. Right, we have that. The total of forces acting in the Y direction is just going to be the normal force minus force due to gravity and that is equal to zero. Since the creates not moving up or down. It means that the normal force is equal to M. G. Now let's use new and second law again. But this time in the X direction is that the sum of all forces in the X direction is equal to mass times acceleration in the X direction. And we need this acceleration for here to find our time. So let's go and solve for that. Right? Well, our total forces in the extraction is just our force due to static friction to our mass times. Acceleration plugging in our values for this. We get that our coefficient of static friction times our normal force is equal to our mass times acceleration the extraction. But we know that our normal force is equal to the force due to gravity which is just MG equal to M A X. Which is super helpful because now we can divide both sides by M. And we get that the acceleration that we are looking for is our coefficient of static friction times our acceleration due to gravity. So let's go ahead and plug those in. We're told that our coefficient of static friction 0.458. The acceleration due to gravity is 9.81. Giving us an acceleration right here giving us an acceleration of 4.49 m per second squared. Great! Now that we have that, let's use another cinematic formula to find our time here. There's a cinematic formula that states that our final velocity is equal to our initial velocity plus acceleration times time subtracting our initial velocity from both sides and dividing by our acceleration on both sides. This yields that our time is equal to our final velocity minus our initial velocity divided by our acceleration. Let's go and plug in our values. Here we have that our final velocity was 15 minus our initial of zero. Divided by our acceleration that we just found of 4.49 gives us the fastest time for the forklift to accelerate of 3.34 seconds corresponding to answer choice B. Thank you all so much for watching. Hope this video helped. We will see you all in the next one