Help Solving Angular Work Problem: 11676 J?

[Punchlinegirl]

A 42.0 kg wheel, essentially a thin hoop with radius 0.90 m, is rotating at 250 rpm. It must be brought to a stop in 10 s. How much work must be done to stop it.

I used the equation delta K = 1/2 I w^2_f - 1/2 I w^2_i = W and got - 11676 J, but this is not right. Help?

 

[xanthym]

A 42.0 kg wheel, essentially a thin hoop with radius 0.90 m, is rotating at 250 rpm. It must be brought to a stop in 10 s. How much work must be done to stop it.

I used the equation delta K = 1/2 I w^2_f - 1/2 I w^2_i = W and got - 11676 J, but this is not right. Help?

From problem statement:
{Hoop Mass} = (42.0 kg)
{Hoop Radius} = (0.9 m)
{Moment of Inertia for Thin Hoop} = M*R^2
{Frequency of Rotation} = f = (250 RPM) = (4.1667 rev/sec)
{Angular Velocity} = ω = 2*π*f = 2*π*(4.1667) = (26.18 radians/sec)

{Rotational Kinetic Energy} = (1/2)*I*ω^2 =
= (1/2)*(M*R^2)*ω^2 =
= (1/2)*(42.0)*{(0.9)^2}*{(26.18)^2} =
= (11659 J)


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[thepatientmental]

Firstly, great job on using the correct equation to solve this problem! However, the negative result you got may be due to the fact that you used the wrong values for the moment of inertia (I) and final angular velocity (w_f).

To calculate the moment of inertia for a thin hoop, we use the formula I = mr^2, where m is the mass and r is the radius. So in this case, the moment of inertia should be I = (42.0 kg)(0.9 m)^2 = 34.02 kgm^2.

Next, we need to convert the given angular velocity of 250 rpm to radians per second. Recall that 1 revolution = 2π radians, so 250 rpm = (250/60) rev/s = (25/6) rev/s = (25/6)(2π) rad/s = 26.18 rad/s.

Now, plugging in the correct values into the equation, we get:

delta K = 1/2 (34.02 kgm^2)(0) - 1/2 (34.02 kgm^2)(26.18 rad/s)^2 = - 23,724.75 J

Since the result is still negative, it means that the work done to stop the wheel is actually 23,724.75 J. This makes sense because in order to stop an object's motion, we need to apply a force in the opposite direction, which requires work to be done.

I hope this helps clarify the confusion and provides the correct solution to the problem. Keep up the good work!