An antique cabinet is in the cab of a pickup truck traveling 20.1 m/s and has a coefficient of static friction of 0.250 between the antique and the cab. In the event of a sudden stop, what is the shortest distance the pickup truck can stop without having the cabinet move around and get damaged?
To determine the shortest distance the pickup truck can stop without having the cabinet move around and get damaged, we need to calculate the maximum deceleration that can be tolerated by the static friction between the cabinet and the cab.
The maximum static frictional force can be calculated using the equation:
Fs = μs * N
where Fs is the maximum static frictional force, μs is the coefficient of static friction, and N is the normal force.
The normal force on the cabinet can be calculated using the equation:
N = m * g
where m is the mass of the cabinet and g is the acceleration due to gravity.
Let's assume the mass of the cabinet is 100 kg.
The normal force can be calculated as follows:
N = m * g
N = 100 kg * 9.8 m/s^2
N = 980 N
Given that the coefficient of static friction is 0.250, we can calculate the maximum static frictional force as follows:
Fs = μs * N
Fs = 0.250 * 980 N
Fs = 245 N
Now, we need to find the deceleration required to stop the pickup truck without the cabinet moving. According to Newton's second law, the force required to decelerate an object is given by:
F = m * a
where F is the force, m is the mass of the pickup truck plus the cabinet, and a is the deceleration.
Let's assume the mass of the pickup truck is 2000 kg.
The force required can be calculated as follows:
F = m * a
245 N = (2000 kg + 100 kg) * a
245 N = 2100 kg * a
Simplifying the equation, we can calculate the deceleration as follows:
a = 245 N / 2100 kg
a ≈ 0.117 m/s^2
Now, we can use the equation of motion to calculate the shortest distance required to stop the pickup truck without the cabinet moving:
v^2 = u^2 + 2a * s
where v is the final velocity (0 m/s), u is the initial velocity (20.1 m/s), a is the deceleration (-0.117 m/s^2), and s is the distance required.
Plugging in the values, we can rearrange the equation to solve for s:
0^2 = (20.1 m/s)^2 + 2 * (-0.117 m/s^2) * s
Simplifying the equation, we get:
0 = 404.01 m^2/s^2 - 0.234 m/s^2 * s
0.234 m/s^2 * s = 404.01 m^2/s^2
s = 404.01 m^2/s^2 / 0.234 m/s^2
s ≈ 1727 m
Therefore, the pickup truck would need to stop within approximately 1727 meters to prevent the antique cabinet from moving around and getting damaged during a sudden stop.