Question

A crate of eggs is located in the middle of the flat bed of a pickup truck as the truck negotiates a curve in the flat road. The curve may be regarded as an arc of a circle of radius 36.1 m. If the coefficient of static friction between crate and truck is 0.570, how fast can the truck be moving without the crate sliding?

Answer 1

Answer:

$v_{max}=14.2\frac{m}{s}$

Explanation:

Hi!

If the crate is not sliding, its trajectory is the arc with 36.1 m radius. Then the crate  has a centripetal acceleration:

$a_c= \frac{v^2}{r} \\r = radius\\v = tangential \; velocity$

The centripetal force acting on the crate is the static friction force between crate and truck. The maximum value of this force is:

$F_{max} = \mu N\\\mu = 0.570=static\;friction \;coefficient\\N =normal\; force$

The normal force has a magnitude equal to the weight of the crate:

$N=mg$

Then the condition for not sliding is:

$F_{centripetal} = M\frac{v^2}{r}<\mu N=\mu Mg\\ v^2<r \mu g = 36.1\;m*0.570*9.8\frac{m}{s^2}= 201.65 \frac{m^2}{s^2}\\ v<14.2\frac{m}{s}$