You absentmindedly leave your book bag on the top of your car. (a) Estimate the safe acceleratio

step 1 Okay, so question 53. You absentmindedly leave your book bag on top of your car. A, estimate the

You absentmindedly leave your book bag on the top of your...

Key Concepts

Simplifying Assumptions in Physical Modeling

In many physics problems, simplifying assumptions—such as neglecting air resistance, assuming a perfectly flat surface, a constant friction coefficient, and rigid-body behavior—are made to isolate the key dynamics of the system. These assumptions allow for tractable mathematical analysis and provide insight into the core relationships between forces in the system.

Force Balance in Accelerated Systems

Establishing a force balance involves setting the maximum available static friction equal to the required force to accelerate an object without slipping. This technique is used to determine the safe acceleration limits and, by extension, the safe speed under certain conditions, by comparing the required force to the threshold provided by friction.

Newton’s Laws of Motion

Newton’s laws, particularly the first law (inertia) and the second law (F = ma), are fundamental in understanding how objects behave when a force is applied. In the context of accelerating vehicles, these laws explain why an object tends to remain at rest and how a net external force (like friction) is needed to accelerate the object along with the vehicle.

Static Friction

Static friction is the force that resists the relative motion between two surfaces in contact that are not already sliding. It is quantified by the product of the coefficient of static friction and the normal force, and it determines the maximum force available to oppose any external force that would otherwise cause slipping.

Inertial Effects in Non-Inertial Frames

When analyzing situations from the perspective of an accelerating vehicle, inertial (or pseudo) forces appear to act on objects. Recognizing these apparent forces is crucial for setting up a correct force balance, ensuring that the net force provided by friction is sufficient to counteract the inertial tendency of an object to remain in its original state of motion.

Transcript

00:01 Okay, so question 53.

00:04 You absentmindedly leave your book bag on top of your car.

00:09 A, estimate the safe acceleration of the car needed for the bag to stay on the roof, describe the assumptions that you've made, and b, estimate the safe speed describing the assumptions that you've made.

00:22 Okay, so let's just assume that the top of the car is modeled as a horizontal surface, and we can draw the book bag.

00:30 Just some block and we'll have a normal force and the weight of the book bag and of course if we've got an acceleration we're going to have force of mass time acceleration is going to be pushing us backwards because we're moving forwards and by newton's third law and a causes us to move backwards and the force of friction obviously prevents this from happening which is why things don't instantly fall off surfaces so for the acceleration side of things for part a, we'll just take a quick look here and we'll do some force balancing.

01:07 So the normal reaction force has to be equal to the weight, so n is equal to m g.

01:13 And for the horizontal forces, ma must equal the coefficient of friction multiplied by the normal reaction force, which as we've already worked out is m g.

01:27 And here the masses cancel, so it doesn't actually matter how much the mass of the book bag is.

01:34 Okay, and now we have to make an assumption.

01:37 So i'm going to write our assumptions in red.

01:41 And our first assumption is that the coefficient of friction between the car and the book bag has got to be some value.

01:48 And so you've got to look something like this up, really, in this sort of question.

01:53 Look for the idea of a fabric sliding against a metal.

01:57 That'll give you some idea, because of course the coefficient of friction is about what two things are sliding against each other.

02:04 A very rough estimate would place a coefficient of friction, say, at 0 .5.

02:09 And so, therefore, a, let's undo that, i'd rather just write the estimations in red and everything else in another colour.

02:19 So acceleration is just going to be equal to our coefficient of friction, which we've estimated multiplied by 9 .8, which is the value of g.

02:28 And so we're going to give that to be a value of 4 .1.

02:33 Meters per second squared.

02:38 Okay.

02:40 And now for the second part, which is estimate a safe speed describing the assumptions we've made.

02:47 This is very interesting because really for a speed we're assuming actually a force is acting due to a speed.

02:56 But of course if velocity is constant, there's no acceleration and therefore no net force.

03:01 But this is an idealized version of the world.

03:05 In the actual reality, of course, as you go faster, you do end up with a force.

03:11 And that's the force of drag.

03:13 So the force of friction has to run up to the point where it can overcome the force of drag.

03:20 And if the force of drag, due to your velocity gets too great, friction will be overcome and the book bag will slide off.

03:31 So, well, the force of friction is just going to be equal to your coefficient of friction, static friction, in this case, multiplied by mg.

03:41 And that's going to be equal to your force of drag...

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