Calculating Power Transferred in a 2-Pulley System

[stackemup]

Homework Statement

From the below information provided, I have to calculate the power which has been transferred to the second pully.

2 pulley system, first pulley has a diameter of 150mm and driven by an electric motor at 250 rev min -1. , second pulley has a diameter of 400mm and is driven by a v-belt, at a load of 200NM. The distance between the two pulleys is 600mm.
Included angle of the pulley groove is 40°. Coefficient of friction is 0.4 between belt and pulley. The belt has an ultimate strength of 8kN.

Homework Equations

Power = Torque x Angluar Velocity

The Attempt at a Solution

From one of the other threads I found the below -

"Driver speed = 250 revs min-1 = 25pi/3 rad s-1 = 26.18 rad s-1
Driven Speed (angular Velocity) = (25pi/3) * (0.075/0.2) = 25pi/8 rad s-1 = 9.817 rad s-1
Power dissipated at pulley 2 = (25pi/8) * 200 = 625*pi W
= 9.817 * 200 = 1963.4 W = 1.9634 kW"

I need to understand how the angular velocity has been calculated, as I am unsure where the figures have come from.

 

[stackemup]

Anyone :)

 

[haruspex]

Are you asking for an explanation of this:

Driven Speed (angular Velocity) = (25pi/3) * (0.075/0.2)

?
If the driving pulley is turning at rate $\omega_1$ and has radius r₁, what is the rate of travel of the belt?
If the driven pulley has radius r₂, what is its angular rate?

 

[stackemup]

I'm unsure on how to calculate the above.

I have found that to calculate the power, the net torque is multiplied by the angular velocity and radius. Is this correct ?

Motor speed = 250 rev min -1

Angular velocity = (250 x 2pi) / 60
= 26.18 rads -1

Is this correct so far?

 

[haruspex]

I'm unsure on how to calculate the above.

If a pulley radius r turns through an angle $\theta$ (in radians), how far does the belt move?
If it does that in time t, what is the average angular velocity ($\omega$) and what is the average belt speed?

Motor speed = 250 rev min -1

Angular velocity = (250 x 2pi) / 60
= 26.18 rads -1

Is this correct so far?

Yes.

 

[stackemup]

Still struggling to understand.

If I take the angular velocity calculated above, I need to multiply this by the torque and effective radius to achieve the power transmitted into the second pulley, correct?

I'm assuming that in the answer previously calculated in the other thread, uses the effective radius of 0.075 (rad 1) / 0.2 (rad 2?) = 0.375, but I need an understanding of why these are used as I cannot find anything through internet sources etc.

The final answer quoted in my original post was calculated by another member so this is irrelevant, I just need an understanding of the full working out process.

 

[stackemup]

Can anyone advise?

 

[haruspex]

If I take the angular velocity calculated above, I need to multiply this by the torque and effective radius to achieve the power transmitted into the second pulley,

No. Check the dimensions:
angular velocity dimension = T^-1
torque dimension = force * distance = ML²T^-2
radius dimension = L
multiply together: ML³T^-3
power dimension = energy/time = ML²T^-3
So what do you think you might have wrong?

uses the effective radius of 0.075 (rad 1) / 0.2 (rad 2?) = 0.375, but I need an understanding of why these are used

I was trying to explain that to you. Please try to answer my questions:

If a pulley radius r turns through an angle θ (in radians), how far does the belt move?
If it does that in time t, what is the average angular velocity (ω) and what is the average belt speed?

 

[stackemup]

I have finally worked it out and now understand it. Can you please review my below answer to find the power transferred if the max tension is halved to 4kN.

F1 = 4000N
F1/F2 = e^(μθ/sin⁡α )
4000/F2 = e^((0.4 x 2.722)/sin⁡20 )
4000/F2 = e^3.1834
4000/F2 =24.1296
F2= 4000/24.1296
F2= 165.7715

T= (F1 - F2 ) r
T= (4000 -165.7715) x 0.075
T=287.5671 Nm

P= Tω
ω= (250 x 2π)/60 = 26.1799

∴ using a torque load of 287.5671 Nm the actual power transmitted:
P= 26.1799 x 287.5671
P= 7.5285 kW

 

[haruspex]

F1 = 4000N

That's the max tension, but at this stage we don't know if that will be reached. Start with the actual load.

e^(μθ/sin⁡α )

Please define all your variables, otherwise I have to guess what you are doing. What are these angles?