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Hanna is pulling an object of 20 kg over a horizontal plane. The force Hanna is exerting makes an angle of 30Â° with the horizontal. The coefficient of sliding friction Î¼, between the object and the plane, is 0.57.

If the object is moving at constant velocity, what is the magnitude of the force provided by Hanna?

First of all, let's represent what is happening in the problem with a simple sketch.

We will need to represent a horizontal surface, an object on it, and indicate that the object is moving at constant velocity, as well as the fact that the object is being pulled by a force that makes 30Â° with the horizontal, and is subject to the force of friction:

By observing the sketch, we notice that the object is subject to 4 forces:

- Hanna's pulling force, F
- the sliding friction force, F
_{f} - the gravitational force, mg
- and the the normal force, N

So, the free-body diagram of the object will look something like this:

We know the angle that the pulling force makes with the horizontal (30Â°), the mass of the object (20 kg), the coefficient of sliding friction (0.57), and that the object is moving at *constant* velocity.

We need to find the magnitude of the pulling force exerted by Hanna.

Î¸ = 30Â°

m = 20 kg

Î¼ = 0.57

v is constant

F = ?

The fact that the object is moving at constant velocity tells us that the object *has no* acceleration (because if it had, it wouldn't be moving at constant velocity):

a = 0

And because the acceleration is zero, the resultant force acting on the object must also be zero (for **Newton's 2 ^{nd} Law**):

R = ma

R = m Ã— 0

R = 0

Knowing the resultant force, we can use the following strategy to find the magnitude of Hanna's pull:

- We find the x and y components of the resultant force, as a sum of the x and y components of all the forces that act on the object.
- Because we already know that R is zero, R
_{x}and R_{y}must also be zero, so we substitute their values in the equations that we found in step 1. - We then use those equations to find F (the pulling force).

We begin by drawing coordinate axes on our free-body diagram and finding the components of all the forces that act on the object:

F_{x} = F cos 30Â°

F_{fx} = âˆ’F_{f}

N_{x} = 0

mg_{x} = 0

F_{y} = F sin 30Â°

F_{fy} = 0

N_{y} = N

mg_{y} = âˆ’mg

Next, we find the x and y components of the resultant force by adding all the x and y components:

x:

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

R_{x} = F cos 30Â° + (âˆ’F_{f}) + 0 + 0

R_{x} = F cos 30Â° âˆ’ F_{f}

y:

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

R_{y} = F sin 30Â° + 0 + N + (âˆ’mg)

R_{y} = F sin 30Â° + N âˆ’ mg

And since

R_{x} = 0

R_{y} = 0

we substitute their values and get

0 = F cos 30Â° âˆ’ F_{f} (1)

0 = F sin 30Â° + N âˆ’ mg (2)

These two equations have three unknowns (F, F_{f} and N).

The number of equations has to be equal to the number of unknowns in order to solve them. Therefore, we should somehow reduce the number of unknowns to *two*.

We know the coefficient of sliding friction Î¼, and the sliding friction force has by definition the magnitude equal to Î¼ multiplied by N:

F_{f} = Î¼N

Which means that we have reduced the number of unknowns to two, and we can now solve the two equations.

By exchanging F_{f} with Î¼N in Eq. (1), we get

0 = F cos 30Â° âˆ’ Î¼N (3)

So, we now have two independent equations (Eq. (2) and Eq. (3)).

First, we solve one of them for one unknown: let's solve Eq. (2) for N:

0 = F sin 30Â° + N âˆ’ mg

F sin 30Â° + N âˆ’ mg = 0

N = mg âˆ’ F sin 30Â°

And substitute N in Eq. (3):

0 = F cos 30Â° âˆ’ Î¼N

0 = F cos 30Â° âˆ’ Î¼ (mg âˆ’ F sin 30Â°)

Finally, we solve this equation for F:

0 = F cos 30Â° âˆ’ Î¼ (mg âˆ’ F sin 30Â°)

F cos 30Â° âˆ’ Î¼ (mg âˆ’ F sin 30Â°) = 0

F cos 30Â° âˆ’ Î¼mg + Î¼F sin 30Â° = 0

F cos 30Â° + Î¼F sin 30Â° = Î¼mg

F (cos 30Â° + Î¼ sin 30Â°) = Î¼mg

F = | Î¼mg |

cos 30Â° + Î¼ sin 30Â° |

F = | (0.57) (20 kg) (9.8 N/kg) |

cos 30Â° + (0.57) (sin 30Â°) |

F = | 112 N |

1.15 |

F = 97 N

Hence, Hanna's pulling force has a magnitude of 97 N.

- Remember that whenever an object is either at
*rest*or moving at*constant velocity*, the resultant force acting on the object is 0. - When you have two independent equations with two unknowns, you can solve them in different ways. Usually we tend to solve one of the two equations for one unknown, and substitute the result in the second equation, so that we end up with an equation that has only one unknown, which we can solve. Once we solved that equation and found the first unknown, we can substitute the found value in the other equation, and find the second unknown.

A light box of 1.3 kg is pulled over a horizontal table with a force that makes an angle of 45Â° with the horizontal. Knowing that the box is moving at *constant* velocity and that the coefficient of sliding friction is 0.80, find the *magnitude* of the pulling force.

F = 8.0 N

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