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 39.6 m. If the coefficient of static friction between crate and truck is 0.630, how fast can the truck be moving without the crate sliding?
ma=F(fr),
mv²/R =μmg,
v=sqrt(μRg) = sqrt(0.63•39.6•9.8)= …
To determine how fast the truck can be moving without the crate sliding, we need to consider the maximum centrifugal force that can be exerted on the crate without exceeding the force of friction.
The centripetal force acting on an object moving in a circular path is given by the equation:
F_c = m * (v^2 / r),
Where F_c is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circular path.
In this case, the crate of eggs will experience a centrifugal force equal to the centripetal force acting on it. Therefore:
F_f = m * (v^2 / r)
Where F_f is the force of friction between the crate and the truck, which acts as the centripetal force.
The maximum force of static friction can be calculated using the equation:
F_f_max = μ * F_n,
Where F_f_max is the maximum force of static friction, μ is the coefficient of static friction, and F_n is the normal force exerted on the crate.
In this scenario, the normal force F_n is equal to the weight of the crate, which can be calculated as:
F_n = m * g,
Where m is the mass of the crate and g is the acceleration due to gravity.
Combining these equations, we can substitute the maximum force of static friction equation into the centripetal force equation:
μ * F_n = m * (v^2 / r),
Simplifying, we can cancel out the mass (m) on both sides of the equation:
μ * g = v^2 / r.
Rearranging the equation to solve for v:
v = sqrt(μ * g * r).
Now we can substitute the given values:
μ = 0.630,
g = 9.8 m/s^2,
r = 39.6 m.
Calculating the final result:
v = sqrt(0.630 * 9.8 * 39.6) = 20.075 m/s.
Therefore, the truck can be moving at a maximum speed of 20.075 m/s without the crate sliding.