A coffee cup on the horizontal dashboard of a car slides forwa...

Question

A coffee cup on the horizontal dashboard of a car slides forward when the driver decelerates from 45 km/h to rest in 3.5 s or less, but not if she decelerates in a longer time. What is the coefficient of static friction between the cup and the dash? Assume the road and the dashboard are level (horizontal).

Accepted Answer

Welcome back. Everyone in this problem. A book is lying on the floor of a train that is moving at 60 kilometers per hour. It only slides forward if the train comes to a stop in not more than five seconds, we want to calculate the coefficient of static friction between the book and the floor of the train. Assuming the floor of the train is perfectly aligned horizontally with the ground outside the train. For our answer twice is a says it's negative 0.3 B 0.03 C 0.3 and D 30. No, let's try to draw a diagram of what's going on here to better understand. So we're talking about a book lying on the floor of a train, ok? And for this book, we know that our train is moving at a speed of 60 kilometers per hour, ok? And since we're gonna try and do some to, to we, we're gonna work with some other values that would be in SI units. Let's change our 60 kilometers per hour before we move any further. So let's put it in SI units which is meters per second and to do that we would multiplied. Ok. Multiplied by 1000, divided by 3600 1000 m in a kilometer, 3600 seconds in an hour. And when we do that, we can rewrite our 60 kilometers per hour as 16 and two thirds. Ok. Let me write that properly. 16 and two thirds meters per second. Ok. So let's just keep that in mind. I just wanted to clear that up before we continue. OK. So we know that in SI units it's moving at a speed moving at a speed of 16 and two thirds meters per second. OK. And we also know we also know that our book has a weight here. We're not given the weight, OK? But we know that it has a weight acting downwards and that weight is opposing the normal force. We're gonna come back to the normal force a bit later. No, the friction force is the force that is exerted on the book that causes the book to decelerate in this case. So if our train is going to stop, then that force that friction force would be acting in the opposite direction. So the direction the train is moving in. So let's call that force fs OK. Now there's a maximum static friction force that can be applied on the book by the floor. And therefore a maximum possible deceleration of the book. And this maximum static friction force is given by the formula frictional force equals mu multiplied by N where MU is a coefficient of static friction and N is the normal, the perpendicular force that's acting at the point of contact. That is now, since our normal is acting against the weight of the book, then we can say then that our fictional force is multiplied by MG. And since we're solving for frictional force, we can rearrange our formula to say that MU equals our frictional force divided by MG. Now, the thing is here, we don't know the mass of the book. OK. And neither do we know the frictional force. However, we know that if the deceleration of the train is greater than the deceleration of the book that is caused by the friction force, then the book would slide forward with respect to the train. So if we're going to figure out this frictional force, let's ask ourselves, what else do we know about force? What else do we know? Well, we also know that based on Newton's second law force is directly proportional to acceleration or in this case, deceleration. So our force would be equal to a mass multiplied by deceleration. Put a negative sign here to show our acceleration is negative or rather that it's decelerating. So our force is going to be equal to oops or force is going to be equal to mass multiplied by acceleration, which tells us then that our friction of force, our coefficient of static friction rather is going to be equal to negative M divided by MG, we can cancel M which means that our coefficient of tactic friction equals negative A divided by G. No, we've been able to rearrange our formula. But again, we know what our gravity is but our acceleration due to gravity, but we don't know our deceleration. So how can we figure that out? Well, again, let's think about what acceleration is. OK. We call that for the least deceleration of the train, the static friction would be maximum and the deceleration of the book would be equal to the deceleration of the train. So if we find that deceleration of the train, then we'd be able to find the acceleration of the book at that least the acceleration, no acceleration. OK. OK. We call that acceleration is equal to the change in speed over time which would be equal to the final speed minus the initial speed divided by the time. OK? And for our train, it had an initial speed of 60 kilometers per hour or 16 and two thirds meters per second, it stopped. So it went to 0 m per second and it did this like our problem says in less than five seconds. So we can take our time to be five seconds. So if our time is five seconds and we have all that information, we can find our acceleration because no solving for air or in this case or deceleration, then it's going to be equal to our final speed zero minus our initial speed 16 and two thirds meters per second. And all of that is going to be divided by the time five seconds. Not when we work that out. When we put that into our calculator, we should get that. Our acceleration equals negative three and a third meters per second squared. So that's our deceleration. OK. As we said, the deceleration for the chain is the same as the deceleration for the book at that point. So we can use that to solve for the static coefficient friction of the book because now that tells us then that mu is going to be equal to negative so negative negative three and a third meters per second squared divided by G acceleration due to gravity which is 9.81 m per second squared. Now, if we put this into our calculator, OK? Put this into our calculator, we should find that we get mu to be equal to 0.3398. If we take it to one significant figure, then me it is going to be approximately equal to 0.3. So that tells us then that the coefficient of static friction between the book and the floor of the train is going to be 0.3. If we look at our answer choices, C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Newton's First Law of Motion

Friction

Deceleration and Acceleration

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