An object with a mass of #200 kg# is hanging from an axle with a radius of #6 cm#. If the wheel attached to the axle has a radius of #32 cm#, how much work would it take to turn the wheel a length equal to the circumference of the axle?

Question

An object with a mass of #200  kg# is hanging from an axle with a radius of #6 cm#.  If the wheel attached to the axle has a radius of #32 cm#, how much work would it take to turn the wheel a length equal to the circumference of the axle?

Nora Arzate · Accepted Answer

The work is #=138.5J#

The load is #L=200gN#

The radius of the axle is #r=0.06m#

The radius of the wheel is #R=0.32m#

The effort is #=FN#

Taking moments about the center of the axle

#F*0.32=200g*0.06#

#F=200g*0.06/0.32=367.5N#

The force is #F=367.5N#

The distance is #d=2pir=2*pi*0.06=(0.12pi)m#

The work is #W=Fd=367.5*0.12pi=138.5J#

Leo Abeyta · Answer

The work done is 138.5 J.

The circumference of the axle is #2*pi*0.06 m#. The circumference of the wheel is #2*pi*0.32 m#. Therefore since the wheel is turned a length equal to the circumference of the axle, the wheel, and also the axle, is rotated a fraction of 1 revolution. The fraction is the ratio, k, of the 2 circumferences #k = (cancel(2)*cancel(pi)*0.06 m)/(cancel(2)*cancel(pi)*0.32 m) = 0.1875# Therefore the object is lifted a distance, #Deltah#, that is #0.1875*2*pi*0.06 m#. By the principle of conservation of energy, the work that was done is equal to the increase in the object's gravitational potential energy, GPE. #DeltaGPE = m*g*Deltah# #DeltaGPE = 200 kg * (9.8 m)/s^2 * 0.1875*2*pi*0.06 m = 138.5 J# This could also be calculated by multiplying the torque by the angular translation. The torque would be equal and opposite the torque that the object's weight applies to the axle. The angular translation would be k2pi radians. #W = Tau*Theta = m*g*r*k*2*pi# #W = 200 kg*9.8 m/s^2*0.06 m*0.1875*2*pi = 138.5 J# I hope this helps, Steve

Julia Aguilar · Answer

undefined To calculate the work required, we first need to find the circumference of the axle:

Circumference of the axle = 2 * π * radius of the axle

Circumference of the axle = 2 * π * 6 cm = 12π cm

Next, we need to find the distance the wheel needs to be turned, which is equal to the circumference of the axle:

Distance to be turned = Circumference of the axle = 12π cm

Now, we can calculate the work done:

Work = Force * Distance

The force exerted by the object hanging from the axle is equal to its weight:

Force = mass * acceleration due to gravity

Force = 200 kg * 9.8 m/s² = 1960 N

Now, we need to convert the distance to meters:

Distance = 12π cm * (1 m / 100 cm) = 0.12π m

Now, we can calculate the work:

Work = Force * Distance

Work = 1960 N * 0.12π m ≈ 737.3 J

Therefore, it would take approximately 737.3 joules of work to turn the wheel a length equal to the circumference of the axle.