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The coefficient of static friction between a box and a ramp is $0.5 .$ The ramp's incline angle is $30^{\circ} .$ If the box is placed at rest on the ramp, the box will do which of the following?(A) Accelerate down the ramp(B) Accelerate briefly down the ramp but then slow down and stop(C) Move with constant velocity down the ramp(D) Not move

Physics 101 Mechanics

Chapter 4

Newton’s Laws

Newton's Laws of Motion

Applying Newton's Laws

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in this question. We have a box on a ramp and the coefficient of static friction is 0.5. Ah, the ramps inclined angle is 30 degrees. And if the box is placed at rest on the ramp were asked to determine what is going to happen to the box, will it accelerate? Will it remain in place, etcetera? So, as usual with forces questions, we should start off with everybody diagram. So we've got our incline here. It's 30 degrees, and perhaps the box is placed up here. We know that the gravitational force will always point straight down, and we've got a normal force that is perpendicular to the surface. And then since the on the other force, um, playing the box in the direction along the ramp is down the ramp, we know that the friction force will have to be directed up the ram. So to go ahead and get started, um, we can first determine whether or not the box will move initially. So static friction, um, is a little bit of a funny thing because it could be anywhere between zero and some maximum value. Um, that is set by the coefficient of static friction and the normal horse so static friction can increase up to this maximum amount. Um, until it is overcome by some applied force. Um, the thing that is acting against the friction force in this case is the gravitational force. So the gravitational force, um, in the direction along the incline is what is trying to pull. So I'll just put that in red here. What is trying to pull the box down the ramp? The friction force here is trying. Thio, stop that from happening. So the question is, you know which is greater the gravitational force or the maximum static friction force. If the gravitational force is bigger, then the gravitational force will win and the box will start to move. If the maximum static friction force is bigger than the static, friction will be able to keep the box in place against gravity. So let's go ahead and try to figure out what each of these two things would be. So the gravitational force here, um, in the direction of the inclined plane Um, let's call that the X direction. So I'm just going to dio quick coordinate system here. So the gravitational force in the extraction is going to be just a component of the total force. And this angle at the top of this triangle here, um, that is going to be 30 degrees, just like the incline, um, plane. So that's always the case there. If you draw this triangle with your gravitational force, um, then the angle at the top of the triangle always be equal to the angle of the inclined plane. So taking a look at that triangle if we want, we can even draw it kind of separately so that it's easier to see. So taking a look at that triangle we can use so Kitona and figure out what the gravitational force in the extraction will be. So this is 30 degrees up here and so we can see that opposite. Over hypotenuse f g X over f g has to be equal. Thio sign 30. And so the gravitational force is equal to F G sign 30 degrees okay. And then we can take a look at the maximum static friction force as already stated. That is, um, U S f n. And in order to figure out what f n has to be, we'll note that F N has to be completely balanced by the F by the y component of F G. So that's to say F N is equal toe f g y. That's because there's no acceleration in that wide direction there. Um, so if there's no acceleration, the net force has to be zero. And that means that the force in the upward direction would have to completely balance out the force in the downward direction. So f n is equal. Thio F g y, and using this same triangle appear we can see that F g y is f g Khost 30. And now we can very easily compare the gravitational and the maximum static friction because we can see that both are just a multiple of F. G. We don't actually know what F G is because of the fact that we don't have the mass of this object. But that's not actually going to matter. So sign 30. If you look at that in the calculator, that's going to be 0.5 and then, um, us is 0.5. Khost, 30 is 0.8 ah six. And so when you multiply those two together, you get 0.43 f g. Okay, so F G X is 0.5 times f g. F s max is 0.43 f g. So we can see that f g x wins. It is bigger than the maximum static friction force, meaning that that gravitational force is going thio overcome the static friction force. Um and the box is going to start to slide down the ramp. Now, you might be wondering at this point, while does it slide down at a constant speed, Um, does it accelerate? So we have to think about what's gonna happen when, um, it starts to slide when it starts to slide. We're going to go from static friction to kinetic friction because the thing is not moving. Um, but the thing about static friction and kinetic friction is that the static friction tends to be more than the maximum value of static. Friction tends to be more than the kinetic friction. So we have since F G X is greater than the max. We know that it's also going to be greater than the kinetic friction because of this relation up here. So what that means is that, um you know, F G X is greater than f k, which means that we're gonna have an F net that is positive down the ramp and the acceleration down the ramp in that extraction is also going to be positive. So this thing is going thio start to move because the static friction force is going to be overcome and it's going to continue to accelerate down the ramp. So that is in correspondence. Thio answer choice A here.

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