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## Homework Statement

My ENTIRE AP physics class is stumped on the way to correctly write the formula to find acceleration for a falling block (mass of "m") attached by a string (tension of "t") to a fixed, rotating pulley (mass of "M", radius of "r"). Our teacher told us that for a block attached on the left side of a pulley (rotating counterclockwise) the tangential acceleration ("a") of the pulley should be positive (+) and for one on the right side (pulley rotating clockwise) a should be negative (-). We cannot come to a decisive conclusion as to what the final equation to find a for each side should be.

m = block mass

M = pulley mass

r = radius of pulley

t = tension of string

a = tangential acceleration (acceleration of the block)

I = Inertia of pulley

A = angular acceleration of pulley

g = Gravity (assume g=10 if needed)

## Homework Equations

**Positive "a" Formula Set:****(+)Formula One: Pulley Rotating CCW**T = rt, T = IA , I = 1/2M(r^2), and A = a/r

therefore... rt = 1/2 M(r^2)(a/r)

(r^2)t = 1/2M(r^2)a

t = 1/2Ma

**(+)Formula Two: Block Falling to the Left**

t = sum of all forces

t = mg + ma

**Negative "a" Formula Set:****(-)Formula One: Pulley Rotating CW**

T = rt, T = IA , I = 1/2M(r^2), and A = -a/r

therefore... rt = 1/2 M(r^2)(-a/r)

(r^2)t = 1/2M(r^2)(-a)

t = -(1/2Ma)

**(-)Formula Two: Block Falling to the Right**

t = sum of all forces

t = mg + [m(-a)]

or

t = mg - ma

(This part is where we some of the confusion begins. We assume t = mg "-" ma since the block is falling to the right and "a" should be negative)

## The Attempt at a Solution

__Positive, CCW, Left__

t = mg + ma

and

t = 1/2Ma

so

1/2Ma = mg + ma

both sides x 2

Ma = 2mg + 2ma

both sides -2ma

Ma - 2ma = 2mg

factor out a

a(M - 2m) = 2mg

both sides / (M-2m)

**a = 2mg / (M - 2m)**

__Negative, CW, Right__

t= mg - ma

and

t = -(1/2Ma)

so

-(1/2Ma) = mg - ma

both sides x-(2)

Ma = -2mg + 2ma

both sides -2ma

Ma - 2ma = -2mg

actor out a

a(M - 2m) = -2mg

both sides / (M - 2m)

**a = -2mg / (M - 2m)**

I've worked these out to the best of my ability and I believe that they are correct. Can someone please either confirm this or correct me

**ASAP**?

Thanks!