Why Is My Calculation for the Time to Drop an Object Incorrect?

[aa_o]

Homework Statement

Homework Equations

F = ma

The Attempt at a Solution

The gravity pulls on m with a force Fg:
Fg = m*g
This force is directly translated into pulling the total mass m+M with an acceleration a:
(m+M)*a = Fg <=> a = Fg / (m+M) = m*g / (m+M)
with this acceleration, we can calculate the time it takes to drop the distance, d:
d = 1/2 * a * t^2 <=> t = sqrt(2*d / a)
This yields a time of t = 1.73 s, but the answers says 1.0 s.

What am doing wrong!?

 

[TSny]

What am doing wrong!?

Your assumption that mg is the force that pulls the total mass to the right is not correct.

Draw a free body diagram for each mass and set up Newton's second law for each mass.

 

[aa_o]

Ahh, the the tension is of course decreasing as the mass, m, is accelerating. But this means that the force on the total mass is actually less than assumed above T = (g-a)*m, giving me an even longer time?
More specifically:
T = Fx = m*(g-a) = (m+M)*a <=> a = m*g / (2*m + M)
with the same solution for t as above

 

[TSny]

How many horizontal tension forces act on the total mass?

Does the vertical acceleration of m equal the horizontal acceleration of the system?

 

[aa_o]

Okay, i see that we acutally have the tension force acting in the horizontal in 4 places, giving us:
a = 4*m*g / (5*m + M).
Is this correct? (It gives the right answer, but it could be a coincidence!).

It seems that i need to practice my FBD skills, especially when it comes to tension forces.

 

[TSny]

There are not 4 horizontal tensions acting on the (m+M) system. Also, have you taken into account that the vertical acceleration of m is different from the horizontal acceleration of the system?

 

[aa_o]

There are not 4 horizontal tensions acting on the (m+M) system. Also, have you taken into account that the vertical acceleration of m is different from the horizontal acceleration of the system?

Eureka! Of course, both the upper and lower horizontal parts of the string contributes to the acceleration, giving a vertical acceleration that's 2 x that of the system. And then we have 2 horizontal tension forces pulling the system. This gives the same result as above, but with the right physical explanation. Thanks a lot!

 

[TSny]

OK.