A truck is traveling at a speed of 33.1 m/s along a level road. A crate is resting on the bed of the truck, and the coefficient of static friction between the crate and the truck bed is 0.632. Determine the shortest distance in which the truck can come to a halt without causing the crate to slip forward relative to the truck.
μmg≥m|a|
=>
|a|≤μg = 0.632g
To determine the shortest distance in which the truck can come to a halt without causing the crate to slip forward relative to the truck, we need to calculate the acceleration of the truck first.
The force that opposes the motion of the truck and crate is the force of static friction. According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the net force is the force of static friction.
The force of static friction can be calculated using the equation:
Fs ≤ μs * N,
where Fs is the force of static friction, μs is the coefficient of static friction, and N is the normal force.
Since the truck is experiencing only one force, the force of friction, the net force acting on it is equal to the force of friction. Therefore, we can write:
Fs = m * a,
where m is the mass of the truck and crate system, and a is the acceleration of the system.
Rearranging the equation and substituting the formula for static friction, we get:
m * a ≤ μs * N.
Since the truck is traveling on a level road, the normal force N is equal to the weight of the truck and crate system, which can be calculated using the equation:
N = m * g,
where g is the acceleration due to gravity.
Substituting this into the earlier equation, we have:
m * a ≤ μs * m * g.
Canceling out mass, we get:
a ≤ μs * g.
Now we can calculate the acceleration by substituting the given values:
a ≤ 0.632 * 9.8 m/s².
a ≤ 6.17 m/s².
Now that we have the acceleration, we can use the kinematic equation to find the shortest distance:
d = v² / (2 * a),
where d is the distance, v is the initial velocity, and a is the acceleration.
Substituting the values, we get:
d = (33.1 m/s)² / (2 * 6.17 m/s²).
d ≈ 282.67 meters.
Therefore, the shortest distance in which the truck can come to a halt without causing the crate to slip forward relative to the truck is approximately 282.67 meters.