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 37.9 m. If the coefficient of static friction between crate and truck is 0.550, how fast can the truck be moving without the crate sliding?

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 37.9 m. If the coefficient of static friction between crate and truck is 0.550, how fast can the truck be moving without the crate sliding?

Answer 1

Oh.

F(along radius) = ma
μmg=mv²/r
μg=v²/r
μgr=v²
v²=(0.550)(9.80)(37.9)

v ~= 14.3m/s

Answer 2

To find the maximum speed at which the truck can move without the crate sliding, we need to consider the forces acting on the crate. The two main forces are the gravitational force pulling the crate downwards and the frictional force between the crate and the truck bed.

Since the crate is located in the middle of the flat bed of the truck, it experiences a centripetal force towards the center of the circular path. This force is provided by the frictional force between the crate and the truck bed.

The maximum frictional force between two surfaces can be found using the equation:

F_friction = coefficient of static friction * normal force

where the normal force is the force exerted by the truck bed on the crate, which is equal to the weight of the crate (mass * gravity).

Since the crate is not sliding, the maximum frictional force must be equal to the centripetal force:

F_friction = centripetal force

The centripetal force can be calculated as:

F_centripetal = mass * (velocity^2 / radius)

In this case, the radius is given as 37.9 m.

Combining the equations, we have:

coefficient of static friction * (mass * gravity) = mass * (velocity^2 / radius)

The mass cancels out from both sides, so we have:

coefficient of static friction * gravity = velocity^2 / radius

Now, we can rearrange the equation to solve for velocity:

velocity^2 = coefficient of static friction * gravity * radius

velocity = sqrt(coefficient of static friction * gravity * radius)

Plugging in the given values:

coefficient of static friction = 0.550
radius = 37.9 m
gravity = 9.8 m/s^2

velocity = sqrt(0.550 * 9.8 * 37.9)

Calculate the square root and we get the final answer:

velocity ≈ 21.69 m/s

Therefore, the truck can move at a maximum speed of approximately 21.69 m/s without the crate sliding.