Pulley, Moment of inertia, and acceleration

[dalitwil]

A mass of .375kg hangs from a string that is wrapped around the circumference of a pulley with the moment of inertia = .0125 kg*m^2 and a radius of .26m. When the mass is released, the mass accelerates downward and the pulley rotates about its axis as the string unwinds. What is the acceleration of the mass??

I have been using a=rF/mr^2, with my F=mg. The correct answer is 6.57m/s^2, but i can't seem to figure out why.

No rush on answering, the question is from a practice exam I am studying.

Thanks guys.

 

[xanthym]

A mass of .375kg hangs from a string that is wrapped around the circumference of a pulley with the moment of inertia = .0125 kg*m^2 and a radius of .26m. When the mass is released, the mass accelerates downward and the pulley rotates about its axis as the string unwinds. What is the acceleration of the mass??
The correct answer is 6.57m/s^2, but i can't seem to figure out why.

From the problem statement:
{String Tension} = S
{Mass of Suspended Entity} = m = (0.375 kg)
{Weight of Mass} = W = (0.375 kg)*(9.81 m/sec^2) = (3.6788 N)
{Cylinder Radius} = R = (0.26 m)
{Cylinder Moment of Inertia} = I = (0.0125 kg*m^2)

For the suspended entity:
{Net Force} = ma =
= W - S
::: ⇒ S = W - ma ::: Eq #1

For the cylinder:
{Net Torque} = Iα = I*a/R =
= S*R
::: ⇒ S = I*a/R^2 ::: Eq #2

Equating Eq #1 and Eq #2:
W - ma = I*a/R^2
::: ⇒ a = W/{m + I/R^2}
::: ⇒ a = (3.6788 N)/{(0.375 kg) + (0.0125 kg*m^2)/(0.26 m)^2}
::: ⇒ a = (6.5702 m/sec^2)


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[thepatientmental]

The equation you are using, a=rF/mr^2, is the correct equation to use in this situation. However, it is important to make sure that all of the units are consistent in order to get the correct answer. In this case, the mass is given in kilograms and the radius is given in meters, but the moment of inertia is given in kg*m^2. To use this equation, we need to convert the moment of inertia to the same units as the mass and radius.

To do this, we can use the equation I=mr^2, where m is the mass and r is the radius. Plugging in the values given, we get I=0.375kg*0.26m^2=0.0975 kg*m^2. Now, we can use this value for the moment of inertia in the equation a=rF/mr^2.

Substituting in the values for F=mg and the converted moment of inertia, we get a=(0.26m*0.375kg*9.8m/s^2)/(0.0975 kg*m^2*0.26m^2)=6.57m/s^2.

So, the reason why you were not getting the correct answer is because the units for the moment of inertia were not consistent with the units for the mass and radius. By converting the moment of inertia to the same units, we get the correct answer of 6.57m/s^2 for the acceleration of the mass.