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Micheal is pulling a mass of 30 kg up an incline which makes an angle of 26Â° with the horizontal.

The mass has an upward acceleration of 2.5 m/s^{2} and the coefficient of kinetic friction between the mass and the incline is 0.58.

What is the magnitude of the pulling force Micheal is exerting?

As usual, we begin by drawing a sketch of what we think is happening.

We draw an incline that makes an angle of 26Â° with the horizontal, a mass on it, and indicate that the mass is pulled upwards, has an acceleration directed upwards, and is subject to the force of friction:

Looking at the sketch, we can infer that 4 forces are acting on the mass:

- Micheal's pull, F
- the friction force, F
_{f} - the force due to gravity, mg
- and the normal force, N

Let's draw a free body diagram of the mass:

This is what we *know*: the mass (30 kg), the angle that the incline makes with the horizontal (26Â°), the acceleration of the mass (2.5 m/s^{2}), and the coefficient of kinetic friction (0.58).

We need to *find* the force exerted by Micheal.

m = 30 kg

Î¸ = 26Â°

a = 2.5 m/s^{2}

Î¼ = 0.58

F = ?

Since we know the mass and the acceleration, we can find the magnitude of the resultant force by applying **Newton's 2 ^{nd} Law**:

R = ma

â†“

R = ma

Now that we determined the resultant force acting on the mass, we can find F using the following strategy:

- We first find the x and y components of the resultant force, R
_{x}and R_{y}, as a sum of the x and y components of all the forces that act on the mass. - We then find the values of R
_{x}and R_{y}by decomposing R, and substitute them in place of R_{x}and R_{y}in the equations that we found in the previous step. - Finally, we use the resulting equations to find F.

Let's begin with the first step of our strategy, by drawing the coordinate axes on our free-body diagram. For convenience, we choose the x axis to be in the direction of motion. Then, we determine the x and y components of all the forces that act on the mass:

F_{x} = F

F_{fx} = âˆ’F_{f}

N_{x} = 0

mg_{x} = âˆ’mg sin 26Â°

F_{y} = 0

F_{fy} = 0

N_{y} = N

mg_{y} = âˆ’mg cos 26Â°

And we find the components of the resultant force as a sum of the components of all the forces:

x:

R_{x} = F_{x} + F_{fx} + N_{x} + mg_{x}

R_{x} = F + (âˆ’F_{f}) + 0 + (âˆ’mg sin 26Â°)

R_{x} = F âˆ’ F_{f} âˆ’ mg sin 26Â° (1)

y:

R_{y} = F_{y} + F_{fy} + N_{y} + mg_{y}

R_{y} = 0 + 0 + N + (âˆ’mg cos 26Â°)

R_{y} = N âˆ’ mg cos 26Â° (2)

The next step is to determine R_{x} and R_{y} by decomposing R:

First of all, R has the same direction as the acceleration of the mass, i.e. the positive x direction:

Therefore,

R_{x} = R = ma

R_{y} = 0

Now, let's substitute these values in Eq. (1) and Eq. (2), respectively:

R_{x} = F âˆ’ F_{f} âˆ’ mg sin 26Â°

R_{y} = N âˆ’ mg cos 26Â°

â†“

ma = F âˆ’ F_{f} âˆ’ mg sin 26Â° (3)

0 = N âˆ’ mg cos 26Â° (4)

We have two equations with 3 unknowns: F, F_{f}, N.

To find F, we need to reduce the number of unknowns to 2.

We can do that by remembering that the magnitude of the friction force is:

F_{f} = Î¼N

We already know the value of the coefficient of kinetic friction Î¼. Therefore, Eq. (3) becomes:

ma = F âˆ’ F_{f} âˆ’ mg sin 26Â°

ma = F âˆ’ Î¼N âˆ’ mg sin 26Â° (5)

At this point, we have two equations, Eq. (4) and Eq. (5), with two unknowns, F and N, so we can easily solve them.

Let's solve Eq. (4) for N:

0 = N âˆ’ mg cos 26Â°

N âˆ’ mg cos 26Â° = 0

N = mg cos 26Â°

And substitute it in Eq. (5):

ma = F âˆ’ Î¼N âˆ’ mg sin 26Â°

ma = F âˆ’ Î¼mg cos 26Â° âˆ’ mg sin 26Â°

ma + Î¼mg cos 26Â° + mg sin 26Â° = F

F = ma + Î¼mg cos 26Â° + mg sin 26Â°

F = m (a + Î¼g cos 26Â° + g sin 26Â°)

F = (30 kg) [2.5 m/s^{2} + (0.58) (9.8 m/s^{2}) (cos 26Â°) + (9.8 m/s^{2}) (sin 26Â°)]

F = (30 kg) (2.5 m/s^{2} + 5.1 m/s^{2} + 4.3 m/s^{2})

F = (30 kg) (11.9 m/s^{2})

F = 360 N

Hence, Micheal is pulling the mass with a force of 360 N.

- Even though it is good practice to find the value of only those unknowns that are required by the problem, sometimes you have to deal with long equations that have lots of variables in them. In that case it is acceptable to find the values of some unknowns that are not asked by the problem, so that you can keep your equations simple and short (e.g. instead of substituting N with mg cos 26Â°, you can just calculate the value of N, and keep it in the equation as such). In this problem though, we didn't do this, and our equations got a little bit complex, but you should feel free to simplify your equations whenever you think it makes sense.

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