A 3.10 kg block is acted upon by a horizontal force F(x) = 24.0 - 9.0x where the force is measured in Newtons if x is measured in meters. The coefficient of kinetic friction between the block and the flat surface is 0.160. The block is initially at x = 0 m.
What distance must the block travel if the net work done on the block by F(x) and friction combined is to be exactly zero?
At what point along the way does the net force on the block reach zero?
To find the distance the block must travel for the net work done to be zero, we need to calculate the work done by the force F(x) and the work done by friction and set them equal to each other.
The work done by the force F(x) is given by the equation:
Work = Force x Distance
Since the force is given by F(x) = 24.0 - 9.0x, we can integrate this equation to find the work done over a distance x:
Work_done_by_F(x) = ∫(24.0 - 9.0x) dx
Evaluating the integral, we get:
Work_done_by_F(x) = 24.0x - 4.5x^2 + C1
Where C1 is the constant of integration.
To find the work done by friction, we use the equation:
Work_done_by_friction = Force_of_friction x Distance
The force of friction can be calculated using the equation:
Force_of_friction = coefficient_of_friction x Normal_force
The normal force is the force perpendicular to the surface and can be calculated as:
Normal_force = mass x gravity
Given that the coefficient of kinetic friction is 0.160, the mass of the block is 3.10 kg, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the normal force:
Normal_force = 3.10 kg x 9.8 m/s^2
Now we can calculate the work done by friction:
Work_done_by_friction = (coefficient_of_friction x Normal_force) x Distance
Work_done_by_friction = (0.160 x (3.10 kg x 9.8 m/s^2)) x Distance
Work_done_by_friction = (0.160 x 30.38 N) x Distance
Work_done_by_friction = 4.86 N x Distance
Now we can set up the equation for total work done by adding the work done by F(x) and the work done by friction:
24.0x - 4.5x^2 + C1 = 4.86 Distance
To find the distance where the net force on the block reaches zero, we need to find the point where F(x) - Force_of_friction = 0.
Substituting the values into the equation, we get:
24.0 - 9.0x - (0.160 x 30.38) = 0
Now we can solve this equation to find the point along the way where the net force on the block reaches zero:
24.0 - 9.0x - 4.86 = 0
-9.0x + 19.14 = 0
-9.0x = -19.14
x = (-19.14) / (-9.0)
x ≈ 2.13 m
Therefore, the distance the block must travel for the net work done to be zero is approximately 2.13 meters. The net force on the block reaches zero at the point x = 2.13 meters.