Car taking a curve, with friction

[Furby]

Homework Statement

A 3500kg truck is carrying an unsecured 400kg box containing glass.
The coefficient of static friction between the box of glass and the truck is 0.50.
The coefficient of rolling friction between the truck's tires and the road is 0.03.
The coefficient of static friction between the truck's tires and the road is 0.95.
The truck comes to an unbanked curve in the road with a 40m radius.
What is the maximum speed the truck can take the curve without damaging the glass, assuming the glass is damaged should the box slide on the truck, or the truck itself should slide during the curve.

Homework Equations

I assume:
a=(v^2)/r
F(friction)=(coefficient)*F(normal)

The Attempt at a Solution

I'm not completely familiar with the concept, but as I understand it, the car assumes strictly static friction (for my purpose) while attempting the curve, thus the coefficient of rolling friction is completely null, I think.

What I'm not certain with is how exactly the box will react with the truck, being unattached. The only lateral force on the truck attempting the curve is it's static friction which would be:
F(static friction of truck to road)=0.95*((3500kg+400kg)*9.8m/s^2)=36309N

Surely the box's mass would combine with the truck's? But in regards to its being damaged by its own sliding, I assume the same static friction applied to the car is also affecting the box:
F(static friction of box to truck)=0.50*(400kg*9.8m/s^2)=1960N

If it holds true the box will undergo the same force, then you could just ignore the whole system of the car, easily recognizing that maximizing it's friction would completely compromise the box's limits of friction, thus just calculating the box's limits:
1960N=(400kg*v^2)/40m
v=14m/s before the box begins to slide, whereas v=19.29m/s before the car begins to roll.

Thus 14m/s is the maximum speed.

Sorry if it sounds confusing. It doesn't sound very sound to me, either. I guess I'm searching for a confirmation of my logic.

 

[Doc Al]

Sounds good to me. A few minor points:

I'm not completely familiar with the concept, but as I understand it, the car assumes strictly static friction (for my purpose) while attempting the curve, thus the coefficient of rolling friction is completely null, I think.

The rolling friction isn't zero--the tires are still rolling--but it's irrelevant for this problem. It's static friction that exerts the sideways force so the truck can make the curve.

1960N=(400kg*v^2)/40m
v=14m/s before the box begins to slide, whereas v=19.29m/s before the car begins to roll.

At v = 19.29m/s the truck begins to slide outward, not roll. (That's a different problem.)

Thus 14m/s is the maximum speed.

Good.

 

[tomcue]

Your analysis is mostly correct. The truck will experience a lateral force due to its static friction with the road, which will also be acting on the box of glass. However, the mass of the box will not significantly affect the maximum speed at which the truck can take the curve without damaging the glass. This is because the force of static friction is dependent on the normal force, which is the weight of the object in question. In this case, the weight of the box is relatively small compared to the weight of the truck and will not significantly affect the normal force and therefore the maximum friction force that can be applied.

Your calculation for the maximum speed of the truck is also correct. The maximum speed is determined by the point at which the lateral force due to static friction reaches the maximum friction force that can be applied (in this case, the force of static friction between the truck and the road). Any higher speed would result in the truck sliding and potentially damaging the glass.

One thing to note is that the coefficient of rolling friction may still play a small role in the truck's ability to take the curve without sliding. While it may be negligible compared to the static friction, it could still slightly affect the maximum speed. However, your calculation using only the coefficient of static friction is a good approximation.

Overall, your logic and calculation are sound and your answer is correct. It is important to consider the different types of friction and how they may affect the system when solving problems like this. Great job!