Torque and Tangential Acceleration

In summary, the conversation discusses the correct formula for finding acceleration of a falling block attached to a rotating pulley. It is determined that the tangential acceleration of the pulley should be positive for a block on the left side and negative for a block on the right side. The correct equations for finding acceleration in both cases are t = mg + ma and t = -2mg / (M-2m), respectively.
  • #1

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:

Left.jpg


(+)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:

Right.jpg


(-)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!
 
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  • #2
In both cases, you are taking t to be positive when there is a tension in the string. For the lower block, the correct equation is always t = mg + ma if you measure a as positive downwards.

It is obvious physically that the lower block moves downwards, therefore the acceleration a, measured positive downwards, must be positive.

Your "negative, CW, right" answer is wrong. The mistake is saying t = mg - ma.
 
  • #3
Thanks, I'll explain this to the rest of my class!
 

1. What is torque?

Torque is a measure of the force that causes an object to rotate around an axis. It is calculated by multiplying the force applied to the object by the distance from the axis of rotation.

2. How is torque related to angular acceleration?

Torque is directly proportional to angular acceleration, meaning that the more torque applied to an object, the greater the angular acceleration will be. This relationship is described by Newton's second law for rotational motion.

3. What is tangential acceleration?

Tangential acceleration is the rate of change of an object's tangential velocity. In other words, it is the acceleration of an object as it moves along a curved path.

4. How is tangential acceleration related to angular acceleration?

Tangential acceleration and angular acceleration are related by the radius of the object's path and its angular velocity. Tangential acceleration is equal to the product of the angular acceleration and the radius of the object's path.

5. How is torque related to tangential acceleration?

Torque and tangential acceleration are related through the moment of inertia and angular acceleration. Torque is equal to the product of the moment of inertia and the angular acceleration of an object.

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