-the coefficient of static friction between the flat bed of the truck and the box is us 0.3. Determine the minimum stopping distance s which the truck can have from a speed of 70 km/h with constant deceleration if the box is not to slip forward
To determine the minimum stopping distance, we'll use the equation for the coefficient of static friction:
fs = μs * N
where fs is the maximum static friction force, μs is the coefficient of static friction, and N is the normal force acting on the box.
The normal force N is equal to the weight of the box, which is given by:
N = m * g
where m is the mass of the box, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the box is not slipping forward, the maximum static friction force fs would be equal to the force required to decelerate the box:
fs = m * a
where a is the deceleration of the truck.
We can also calculate the initial velocity v0 of the truck and convert it to meters per second:
v0 = 70 km/h = (70 * 1000) / (60 * 60) m/s
Now, rearranging the equations:
fs = μs * N
m * a = μs * m * g
Canceling the mass m on both sides:
a = μs * g
Finally, the minimum stopping distance s can be calculated using the equation for uniformly accelerated motion:
v^2 = v0^2 - 2 * a * s
where v is the final velocity, which is zero in this case.
Plugging in the values:
0 = (70 * 1000) / (60 * 60)^2 - 2 * (0.3) * 9.8 * s
Simplifying:
0 = 19.44 - 58.8 * s
Rearranging the equation:
58.8 * s = 19.44
s = 19.44 / 58.8
Calculating:
s ≈ 0.3306 meters
Therefore, the minimum stopping distance the truck can have is approximately 0.3306 meters.
To determine the minimum stopping distance of the truck, we need to consider the forces acting on the box.
The force of static friction is what keeps the box from slipping forward. It 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 acting on the box.
The normal force is equal to the weight of the box, which can be calculated as:
N = m * g
Where:
m is the mass of the box, and
g is the acceleration due to gravity.
The force of friction is also equal to the mass of the box multiplied by its acceleration:
F = m * a
Since the box is not slipping forward, the force of static friction is equal to the force of friction:
Fs = F
We can rearrange the equation to solve for acceleration:
a = Fs / m
The deceleration of the truck is equal to the acceleration of the box, but with opposite direction:
a = -deceleration
So we have:
-deceleration = Fs / m
Using the equation of motion:
v^2 = u^2 + 2as
Where:
v is the final velocity (zero in this case),
u is the initial velocity (70 km/h),
a is the acceleration (deceleration), and
s is the stopping distance.
We can rearrange this equation to solve for the stopping distance:
s = (v^2 - u^2) / (2 * a)
Substituting -deceleration for a, we get:
s = (v^2 - u^2) / (2 * -deceleration)
Now we can substitute Fs / m for -deceleration:
s = (v^2 - u^2) / (2 * Fs / m)
Finally, we substitute the equation for Fs:
s = (v^2 - u^2) / (2 * μs * N / m)
Since N = m * g, we can simplify further:
s = (v^2 - u^2) / (2 * μs * m * g / m)
s = (v^2 - u^2) / (2 * μs * g)
Plugging in the given values, we can now calculate the minimum stopping distance.