QUIZ 1 AND 2, PROBLEM 3

A block of mass M is pulled with a rope that exerts a force of a magnitude F, causing it to slide over horizontal ground with kinetic coefficient of friction μ (use "mu" when you input your answer). The force is applied at angle θ (use "theta" when you input your answer) above the horizontal shown below.

1) Resolve F⃗ into components using the coordinate system shown.

Fx,Fy=
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μ⋅N
,

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m⋅g
2) Find the sum of the force components in the y-direction acting on the mass (in terms of N,M,g,F, where N is the magnitude of the normal force exerted by the surface on the box). Do not evaluate your expression - that is, do not get a number. Give a formula instead.

∑Fy=
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3) Find the normal force of the ground acting on the mass.

N=
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m⋅g−F⋅sin(θ)
4) Find the net force in the x-direction acting on the mass (in terms of mu, F, M, g, theta):

∑Fx=
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F⋅cos(θ)−μ⋅(F⋅sin(θ)+N−M⋅g)
5) Find the acceleration of the mass in the x-direction (in terms of mu, F, M, g, theta).

a=
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F⋅cos(θ)−μ⋅(F⋅sin(θ)+N−M⋅g)M
6) The block has a normal force N, a friction force Ff, a gravitational force W, and F acting on it. Which of these relationships about the magnitudes of these forces is true when F is angled as shown and is just strong enough to keep the block moving at a constant speed along the ground?

F=Ff and N=W
F=Ff and N>W

F>Ff and N<W

F>Ff and N=W

F>Ff and N>W

F<Ff and N=W

none of the above

To answer these questions, we will need to break down the given information and use some basic principles of physics. Let's go through each question one by one:

1) Resolve F⃗ into components using the coordinate system shown.

To resolve the force F⃗ into its x and y components, we can use trigonometry. The x-component is given by Fx = F * cos(θ), and the y-component is given by Fy = F * sin(θ). Therefore, the components of F⃗ are:

Fx = F * cos(θ)
Fy = F * sin(θ)

2) Find the sum of the force components in the y-direction acting on the mass.

In this case, the sum of the force components in the y-direction is the net force acting vertically on the mass. This is equal to the weight of the object minus the vertical component of the applied force and the normal force acting on the object. Therefore, the formula for the sum of the force components in the y-direction is:

∑Fy = M * g - Fy - N

3) Find the normal force of the ground acting on the mass.

The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is equal to the weight of the object minus the vertical component of the applied force. Therefore, the formula for the normal force is:

N = M * g - F * sin(θ)

4) Find the net force in the x-direction acting on the mass.

The net force in the x-direction is the force responsible for the acceleration of the mass horizontally. It is equal to the horizontal component of the applied force minus the force of friction acting on the mass. Therefore, the formula for the net force in the x-direction is:

∑Fx = Fx - μ * (F * sin(θ) + N - M * g)

5) Find the acceleration of the mass in the x-direction.

The acceleration of the mass in the x-direction is directly proportional to the net force in the x-direction and inversely proportional to the mass of the object. Therefore, the formula for the acceleration is:

a = (∑Fx) / M

6) Determine the relationship between the magnitudes of the forces.

When the applied force is angled as shown and is just strong enough to keep the block moving at a constant speed along the ground, the magnitude of the applied force (F) would be equal to the magnitude of the frictional force (Ff), and the magnitude of the normal force (N) would be equal to the weight of the object (W). Therefore, the relationship is:

F = Ff and N = W

I hope this helps! Let me know if you have any further questions.

To resolve F⃗ into components using the coordinate system shown:

Fx = F * cos(θ)
Fy = F * sin(θ)

To find the sum of the force components in the y-direction acting on the mass:

∑Fy = Fy + μ * N

To find the normal force of the ground acting on the mass:

N = M * g - F * sin(θ)

To find the net force in the x-direction acting on the mass:

∑Fx = Fx - μ * (F * sin(θ) + N - M * g)

To find the acceleration of the mass in the x-direction:

a = (∑Fx) / M

As for the relationship between the forces, when F is angled as shown and is just strong enough to keep the block moving at a constant speed along the ground:

F = Ff and N = W.