An object with a mass of #5 kg# is hanging from an axle with a radius of #24 cm#. If the wheel attached to the axle has a radius of #12 cm#, how much work would it take to turn the wheel #3 cm#?

The work required to turn the wheel by 3 cm would be approximately 90.48 Joules.

To calculate the work required to turn the wheel, you can use the formula for work done in rotating an object:

[ \text{Work} = \text{Torque} \times \text{Angle} ]

[ \text{Torque} = \text{Force} \times \text{Radius} ]

[ \text{Angle} = \frac{\text{Distance}}{\text{Radius}} ]

[ \text{Work} = (\text{Force} \times \text{Radius}) \times \frac{\text{Distance}}{\text{Radius}} ]

Given that the mass is hanging from the axle, we can calculate the force exerted by the mass using ( F = mg ), where ( m = 5 ) kg and ( g = 9.8 , \text{m/s}^2 ).

[ F = 5 , \text{kg} \times 9.8 , \text{m/s}^2 ] [ F = 49 , \text{N} ]

Now, calculate the torque exerted by this force on the wheel attached to the axle, which has a radius of 12 cm.

[ \text{Torque} = 49 , \text{N} \times 0.12 , \text{m} ] [ \text{Torque} = 5.88 , \text{Nm} ]

The angle through which the wheel is turned is given by the formula ( \frac{\text{Distance}}{\text{Radius}} ), where the distance is 3 cm and the radius of the wheel is 12 cm.

[ \text{Angle} = \frac{0.03 , \text{m}}{0.12 , \text{m}} ] [ \text{Angle} = 0.25 , \text{radians} ]

Now, plug these values into the work formula:

[ \text{Work} = 5.88 , \text{Nm} \times 0.25 , \text{radians} ] [ \text{Work} = 1.47 , \text{J} ]

Therefore, it would take approximately 1.47 Joules of work to turn the wheel by 3 cm.