A block of mass m is being dragged across the floor by a force F. The coefficient of friction between the block and floor is u. Show that if the block is being accelerated by the force F from rest for a time t, the speed of the block at time t can be found from: Vf= (F/m - ug)t
To calculate the speed of the block at time t, we will use Newton's second law of motion and the equations of motion for a constant acceleration.
First, let's analyze the forces acting on the block. There are two forces in play: the applied force F and the force of friction between the block and the floor. The force of friction can be given by the equation:
Frictional force = coefficient of friction (u) * normal force
Since the block is being dragged across the floor, the normal force is equal to the gravitational force acting on the block:
Normal force = mass (m) * gravity (g)
Now, let's consider the net force acting on the block in the horizontal direction:
Net force = Applied force - Frictional force
Using the equations mentioned above, we can substitute the respective values:
Net force = F - (u * m * g)
According to Newton's second law of motion, the net force is equal to mass times acceleration:
Net force = mass (m) * acceleration (a)
Setting these two equations equal to each other, we get:
F - (u * m * g) = m * a
Now, we need to find the acceleration (a) based on the applied force and solve for it. Since we know the block is at rest initially, we can use the kinematic equation:
Vf = Vi + a * t
Where:
Vf is the final velocity at time t (what we want to find)
Vi is the initial velocity (which is zero because the block is at rest)
a is the acceleration (which we can solve for using the equation above)
t is the time.
Rearranging the equation, we have:
a = (Vf - Vi) / t
Substituting the initial velocity (Vi = 0), we have:
a = Vf / t
Now, let's substitute this value of acceleration (a) back into the equation from Newton's second law:
F - (u * m * g) = m * (Vf / t)
Now, we can solve for the final velocity Vf:
Vf = (F - (u * m * g)) * t / m
Finally, rearranging the equation, we get:
Vf = F/m - (u * g) * t
Thus, we have derived the equation for the speed of the block at time t, which is:
Vf = (F/m - ug) * t
where:
Vf is the final velocity at time t
F is the applied force
m is the mass of the block
u is the coefficient of friction between the block and the floor
g is the acceleration due to gravity.