A truck moving at 100 km /h carries a steel girder that rest on its wooden floor. what is the minimum time in which the truck can come to a stop without the girder moving forward ? The coefficient of static friction between steel and wood is 0.5.
Well, well, well, looks like we have a truck with a delicate girder situation here. Alright, let's put on our thinking caps and calculate away!
To find the minimum time for the truck to come to a stop without the girder moving forward, we need to consider the forces at play. In this case, it's the force of friction that's preventing the girder from sliding.
The force of friction between the steel girder and the wooden floor can be calculated using the formula: frictional force = coefficient of static friction * normal force.
Normal force is the force exerted perpendicular to the surface, which in this case is the weight of the girder. Since the weight of the girder is not provided, let's simply assume it to be 1, just for fun.
So, the frictional force = 0.5 * 1 = 0.5 N.
Now, to bring the truck to a stop, we need to apply a net force equal to the frictional force. Using Newton's second law (force = mass * acceleration), we can calculate the acceleration required.
Since the mass of the girder is not given either, let's imagine a ridiculously light girder and assume its mass to be 1 gram, just to keep things light-hearted.
So, for the truck to stop, we need a net force of 0.5 N. And force = mass * acceleration, which means the acceleration required is 0.5 m/s^2.
Now, we can use the kinematic equation: final velocity = initial velocity + (acceleration * time).
The initial velocity is 100 km/h, which is equivalent to 27.8 m/s.
Assuming the final velocity is 0 m/s (since the truck comes to a stop), we can rearrange the equation to solve for time:
0 = 27.8 + (0.5 * time)
-27.8 = 0.5 * time
time = -27.8 / 0.5
If we plug the numbers into our equation, we find that the clownishly precise minimum time for the truck to come to a stop without the girder moving forward is approximately -55.6 seconds.
However, please note that this calculation assumes ideal conditions and neglects factors such as air resistance, tire grip, and the actual weight and mass of the girder. So, take it with a pinch of clownish humor!
To solve this problem, we need to find the minimum time in which the truck can come to a stop without the girder moving forward.
Let's consider the forces acting on the girder:
1. The force of gravity acting vertically downward on the girder.
2. The normal force exerted by the wood floor on the girder, which acts vertically upward.
3. The force of static friction exerted by the wood floor on the girder, which acts horizontally backward to prevent the girder from moving forward.
At the minimum, the force of static friction should be equal to the force of gravity acting on the girder in the horizontal direction to prevent it from moving forward. The force of static friction can be calculated using the equation:
frictional force (static) = coefficient of static friction * normal force
In this case, the normal force is equal to the weight of the girder, which can be calculated using the equation:
weight = mass * acceleration due to gravity
As the truck is at rest, the force of static friction is equal to the force of gravity acting on the girder. The truck will come to a stop when the net force acting on the girder becomes zero. The net force can be calculated using the equation:
net force = force of static friction - force of gravity
At rest, the truck has zero acceleration, and the net force is equal to zero. Setting the net force equation to zero, we get:
force of static friction - force of gravity = 0
Substituting the equations for the force of static friction and the force of gravity, we get:
(coefficient of static friction * normal force) - (mass * acceleration due to gravity) = 0
Simplifying the equation, we have:
(coefficent of static friction * mass * acceleration due to gravity) - (mass * acceleration due to gravity) = 0
Factoring out the common term (mass * acceleration due to gravity), we get:
(mass * acceleration due to gravity) * (coefficient of static friction - 1) = 0
Since mass and acceleration due to gravity are positive, we can ignore this term. Therefore, the equation becomes:
coefficient of static friction - 1 = 0
Solving for the coefficient of static friction, we have:
coefficient of static friction = 1
However, given that the coefficient of static friction between steel and wood is 0.5, the truck cannot come to a stop without the girder moving forward at the given coefficient of static friction.
To determine the minimum time in which the truck can come to a stop without the girder moving forward, we need to consider the forces acting on the girder.
The girder will experience two main forces: the force of friction and the force due to inertia.
The force of friction can be calculated using the equation:
F_friction = coefficient of static friction * normal force
In this case, the normal force acting on the girder is equal to its weight, which can be calculated using the equation:
Weight = mass * gravity
The force due to inertia is given by Newton's second law of motion:
F_inertia = mass * acceleration
In this case, the acceleration is the deceleration of the truck, which can be calculated using the equation:
Deceleration = final velocity - initial velocity / time
Since we want to find the minimum time in which the truck can come to a stop without the girder moving forward, we need to determine the maximum deceleration that can be achieved without the girder moving.
The maximum deceleration without the girder moving can be achieved by making the force of friction equal to the force due to inertia:
F_friction = F_inertia
Let's calculate the force of friction:
F_friction = 0.5 * (mass * gravity)
And the force due to inertia:
F_inertia = mass * (final velocity - initial velocity) / time
Equating these two forces, we can solve for the time:
0.5 * (mass * gravity) = mass * (final velocity - initial velocity) / time
Simplifying the equation:
0.5 * gravity = (final velocity - initial velocity) / time
Rearranging the equation to solve for time:
time = (final velocity - initial velocity) / (0.5 * gravity)
Since we know the initial velocity is 100 km/h and the final velocity is 0 km/h (since the truck comes to a stop), and the acceleration due to gravity is 9.8 m/s^2, we can plug in these values:
time = (0 - 100 km/h) / (0.5 * 9.8 m/s^2)
Converting 100 km/h to m/s:
time = (-27.78 m/s) / (0.5 * 9.8 m/s^2)
Calculating the time:
time = -5.66 s
Note: The negative sign indicates that the acceleration is in the opposite direction of the initial velocity. It means the truck is decelerating.
So, the minimum time in which the truck can come to a stop without the girder moving forward is approximately 5.66 seconds.
vf=vi+at
t= vi/a= vi/(force/mass)=-vi*mass/mass*mu= vi/mu=change km/hr to m/s, and you have it.