The Acceleration of a Block on an Incline Plane: Formula and Minimum Ratio

[tony873004]

Can someone tell me if I did this right?

An object of mass m is hung from a rope that passes over a pulley at the top of a ramp and is attached to a block of mass M.

(a) Assume m is large enough that the block accelerates up the ramp. Find an expression for the block’s acceleration.

(b) From your result for part (a), determine the minimum ratio m/M such that the block, once moving, accelerates up the ramp.

(the diagram is simple, M is on the incline plane and m is hanging straight down)

$$a = \frac{Force}{mass}$$

$$a=\frac{Mg sin\theta + \mu Mg cos \theta -mg}{M}$$ (answer for part A)

Set the force equal to 0
$$Mg sin \theta + \mu Mg cos \theta - mg = 0$$
$$Mg sin \theta + \mu Mg cos \theta = mg$$

factor out Mg
$$Mg(sin \theta + \mu cos \theta) = mg$$

the g's cancel

$$m = M(sin \theta + \mu cos \theta)$$

$$m/M = sin \theta + \mu cos \theta$$

 

[dextercioby]

AWKWARDLY,the answer to poin b) is correct (the minimum value is the one u found),but the acceleration you find is incorrect,assuming the rope is inextensible (the same tension in it in every point)...

How about correcting your mistake?

Daniel.

 

[tony873004]

Thanks, Dex.
I'm guessing that I should have M+m in the denominator of my first answer instead of simply M. They must accelerate at the same rate so should be treated as a single mass.

By awkardly, did you mean because I had to get the first part to get the second part?

 

[dextercioby]

Yes,exactly,you got the point perfectly.Well done!

Daniel.

 

[tony873004]

Thanks, Dextercioby. I'd have turned it in as is if you didn't make me go back and look!

 

[dextercioby]

Good thing that u decided to check by posting it here.

Daniel.

 

[tony873004]

Good thing that u decided to check by posting it here.

Daniel.

You couldn't be more right. We had a mid term today, and this exact problem, but with specific numbers, was on the exam. I'd have missed it had we not gone through it last night!