Mass Needed to Accelerate a Box Along a Pulley

[NamikazeBurst]

Homework Statement

Okay, so I am given an acceleration that I need to accelerate a box-thingy, along with a weight inside. Friction is included. I need to come up with a mass that is connected through a pulley to the box. This mass has to accelerate the box at the given acceleration.

I am not asking for the exact mass I need for my need acceleration. I just want to know how to get that mass.

Homework Equations

F_net = ma
##F_g = m_g##
##F_n = F_g##

The Attempt at a Solution

I am having trouble figuring out what mass I need. I know the 'driving force' is ##F_g## (mass I need * 9.8). The resisting forces I believe are the ##F_g## of the box (box total mass * 9.8) and the friction force (I am told I don't need the coefficient).

To get the friction force, I pulled the box with a Newton Spring Scale, and subtracted the net force I want (F=ma) from the pulling force. My equation for acceleration ended up being:

a = (m_Bg - m_Ag - F_f) / m_T

Where the m_Bg is the force from the unknown mass, m_Ag is the force of gravity on the box, F_f is the friction force, and m_T is total mass of the system.

Plugging my numbers in after rearranging for m_B, I ended up getting a small mass (~0.5kg), and I do not believe that would accelerate the box to the rate I need.

What did I do wrong? Is the one force (F_g of box) supposed to be m_Agsinθ (where θ = 0 degrees)? Was the friction force wrong? Did I completely mess up? Please help, and thanks!

Diagram for reference:

Diagram

 

[Dr.D]

Try starting off with FBDs for both bodies, and from that, write the equations of motion.

 

[NamikazeBurst]

Try starting off with FBDs for both bodies, and from that, write the equations of motion.

Okay. So for the box, the force responsible for its acceleration is the tension, and it is reduced by friction for the net force in x. For the mass, it accelerates due to gravity, so mg, and the net force is mg less the tension force.

The equation for the tension force is:
F_t = m₂(g-a)
Mass 2 is the mass I need to find.

The equation for the acceleration of the box is:
a = F_t - F_f / m₁.

Subbing in the the equation for tension gives:
a = (m₂(g-a) - F_f) / m₁

Rearrange for m₂ and I get:
m₂ = (m₁a + F_f) / (g - a)

Is this correct? If so, I still need the force due to friction. My teacher said I didn't need the kinetic coefficient, so there has to be a way to just calculate friction. If I calculate the net force I need to accelerate it at my given acceleration using the mass of the box, and I use the spring scale to calculate the force of kinetic friction by moving the box at a constant speed, would the addition of them be the force I need to achieve the correct acceleration? Doing that gives me a pretty small mass (~0.8kg), and that doesn't seem right to me.

 

[Dr.D]

I don't see any FBDs.

 

[NamikazeBurst]

I don't see any FBDs.

My first little paragraph was based off of FBDs I drew. But, here they are in internet-form.

FBDs

Edit: Wait... if it took 5N to move the box at constant speed, the friction was 5N. That gives me a mass of 1.436kg instead.

 

[Dr.D]

Your FBDs show two Fg values; you need a different notation that will not confuse you or anyone else.

 

[NamikazeBurst]

Your FBDs show two Fg values; you need a different notation that will not confuse you or anyone else.

The Fg on Mass B is mBg, while Fg on the box cancels out with the normal force on the box, so the only Fg involved is FgB. Ignoring that, is this equation correct?
m2 = (m1a + Ff) / (g - a) with friction being the force that the Newton Spring Scale showed when pulling at a constant speed?