A 2.30 kg block is acted upon by a horizontal force F(x) = 30.0 - 10.0x 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 0.170. 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?

Well, well, well, looks like this block has some work to do! To find the distance it needs to travel, we need to find the point where the net work is zero.

First, let's find the work done by the force F(x). The work done by a variable force is given by the integral of that force with respect to distance. In this case, F(x) is equal to (30.0 - 10.0x), so the work done by this force is:

W1 = ∫(30.0 - 10.0x) dx

Integrating this expression will give us the work done by the force F(x). After integrating, substitute the limits of integration from x=0 to x=d (where d is the distance the block needs to travel). Remember, the work done by a force with respect to distance is equal to the negative of the change in potential energy. So, the work done by F(x) is:

W1 = -[15.0x - 5.0x^2] from x=0 to x=d

Next, let's consider the work done by friction. The frictional force is the coefficient of kinetic friction (0.170) times the normal force (which is equal to the weight of the block, mg). The work done by friction is the force of friction times the distance, which is -μmgd (since the frictional force is acting opposite to the motion).

Now, we want the net work (the sum of the work done by F(x) and the work done by friction) to be zero. So, we have:

W1 + Wfriction = 0

Substituting the expressions for W1 and Wfriction:

-[15.0d - 5.0d^2] - (0.170)(2.30 kg)(9.8 m/s^2)d = 0

Now, what we need to do is solve this little equation to find the distance d. You can grab your favorite calculator and tackle it yourself, or you can just let me do all the math and tell you the answer! It's all up to you!

To find the distance the block must travel for the net work done on it to be zero, we need to find the point at which the work done by the applied force F(x) equals the work done by friction.

Let's break down the problem step by step:

Step 1: Find the equation for the frictional force.
The frictional force can be calculated using the equation:
Frictional Force = coefficient of friction * Normal Force

The normal force is equal to the weight of the block, which is calculated as:
Weight = mass * acceleration due to gravity

Given that the mass of the block (m) is 2.30 kg and the acceleration due to gravity (g) is approximately 9.8 m/s^2, we can calculate the weight of the block:
Weight = 2.30 kg * 9.8 m/s^2 = 22.54 N

Now we can find the frictional force:
Frictional Force = 0.170 * 22.54 N = 3.83 N

Step 2: Find the work done by the applied force F(x).
The work done by a force can be calculated using the equation:
Work = Force * Distance

The applied force F(x) is given by the equation: F(x) = 30.0 - 10.0x

To find the work done by this force, we need to find the distance over which the force acts. This distance will be denoted as x.

Work done by the applied force = (30.0 - 10.0x) * x

Step 3: Set the work done by the applied force equal to the work done by friction.
To find the distance at which the net work done is zero, we set the work done by the applied force equal to the work done by friction:

(30.0 - 10.0x) * x = 3.83

Simplifying this equation will give us the value of x, which is the distance the block must travel:

30.0x - 10.0x^2 = 3.83

Rearranging the equation:

10.0x^2 - 30.0x + 3.83 = 0

Now we can solve the quadratic equation to find the value of x.

To find the distance the block must travel, we need to determine the point where the net work done on the block by the force F(x) and friction is zero.

Let's start by breaking down the problem into its components.

1. Work done by the force F(x):
The work done by a force is given by the formula W = F * d * cos(theta), where F is the magnitude of the force, d is the displacement, and theta is the angle between the force and the displacement.

In this case, F(x) = 30.0 - 10.0x represents the force acting on the block. The displacement, d, is the distance the block travels.

Since the force F(x) acts horizontally, the angle between the force and displacement is 0 degrees. Therefore, the work done by the force is Wf = (30 - 10x) * d.

2. Work done by friction:
The work done by friction can be calculated using the formula Wfri = -μk * N * d, where μk is the coefficient of kinetic friction, N is the normal force, and d is the displacement.

The normal force, N, is equal to the weight of the block, which is given by N = mg, where m is the mass of the block and g is the acceleration due to gravity.

Given that the mass of the block, m, is 2.30 kg and the acceleration due to gravity, g, is approximately 9.8 m/s^2, we can calculate the normal force as N = 2.30 kg * 9.8 m/s^2.

Now we can substitute these values into the formula for the work done by friction to get Wfri = -0.170 * N * d.

3. Net work done:
The net work done on the block is the sum of the work done by the force and the work done by friction. Since the net work done is zero, we can set Wf + Wfri = 0 and solve for the displacement d.

(30 - 10x) * d + (-0.170 * N * d) = 0

Substituting the value of N and rearranging the equation, we get:

(30 - 10x) * d = 0.170 * N * d

Now we can cancel out the 'd' term from both sides of the equation:

30 - 10x = 0.170 * N

Solving for x:

-10x = 0.170 * N - 30

x = (30 - 0.170 * N) / 10

Finally, we substitute the values of N and solve for x:

x = (30 - 0.170 * 2.30 kg * 9.8 m/s^2) / 10

After calculating the value of x, we can determine the distance the block must travel by multiplying it by -1 since the displacement is in the opposite direction of the force F(x).

F(x) = 30 - 10x = 0

10x = 30
X = 3 Meters.