An object with a mass of #300 kg# is hanging from an axle with a radius of #6 cm#. If the wheel attached to the axle has a radius of #12 cm#, how much work would it take to turn the wheel a length equal to the circumference of the axle?
The work is
The load L=(300g)N#
Radius of axle Radius of wheel Taking moments about the center of the axle The distance is The work done is
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To find the work required to turn the wheel a length equal to the circumference of the axle, we can use the formula for work done in rotating an object:
[ \text{Work} = \text{Torque} \times \text{Angle} ]
Where:
- Torque (( \tau )) is the force applied perpendicular to the lever arm times the lever arm length
- Angle (( \theta )) is the angle through which the torque is applied
The torque (( \tau )) can be calculated as the product of the force (weight of the object) and the lever arm length (radius of the wheel).
The angle (( \theta )) is given by the distance traveled divided by the radius of the wheel.
First, let's find the torque:
[ \text{Torque} = \text{Force} \times \text{Lever arm length} ]
[ \text{Force} = \text{Mass} \times \text{Gravity} ] [ \text{Force} = 300 , \text{kg} \times 9.81 , \text{m/s}^2 ]
Then, let's find the angle:
[ \text{Angle} = \frac{\text{Distance traveled}}{\text{Radius of the wheel}} ] [ \text{Distance traveled} = 2 \pi \times \text{Radius of the axle} ]
Now, we can calculate the work:
[ \text{Work} = (\text{Force} \times \text{Lever arm length}) \times \frac{\text{Distance traveled}}{\text{Radius of the wheel}} ]
[ \text{Work} = (\text{Force} \times 0.06 , \text{m}) \times \frac{2 \pi \times 0.06 , \text{m}}{0.12 , \text{m}} ]
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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