## Mani_Jha 4 years ago Why ain't the equations of rotational motion like v=w/r dimensionally correct?

1. JamesJ

The units of velocity v is m/s Units of angular velocity is 1/s Units of distance r is m and hence the units of 1/r is 1/m Therefore the units of w/r is 1/s . 1/m = 1/(ms) which is not the units of v: m/s

2. Mani_Jha

Isnt the unit of angular velocity radians/s? And, if they are not dimensionally correct, why do we still use them?

3. JamesJ

Ah, yes, but radians are actually unitless; they are just real numbers. That's why equations like $C = 2\pi r$ are dimensionally correct: $$2\pi$$ radians times $$r$$ meters gives a circumference of $$C$$ meters.

4. Mani_Jha

Thank you very much. But why do we still use these equations despite them being dimensionally wrong?

5. JamesJ

But it's not. The equation is v = wr.

6. JamesJ

Or w = v/r

7. anonymous

a dimensionally correctr equation may not be a correct eqn in physics like that a dimensionally incorrect eqn can be correct dimensional analysis is an experimental concept it does not account for actual correctness of any equation

8. AravindG

a dimensionally wrong eqn is always wrong!! i disagree salini

9. anonymous

sheri aikote njan biology padikan pokunzhu bye nalle velya exam indu eniku

10. AravindG

k bst of luck

11. Mani_Jha

Actually i got this question, while sitting in my car and observing closely the speedometer and the meter showing angular speed(in revolutions/min). I guessed that by noting the readings on the two meters, and by dividing them, I would know the radius of the wheels of my car, all by sitting inside my car! But I had to convert to the appropriate units But now its all clear. Thank u guys!

12. anonymous

There are a few more things to consider. First, the gear ratio in the transmission. Second the gear ratio in the differential. Lastly, if you have an automatic transmission, ensure that the torque converter is locked. Without these considerations you'll find that you're tires are either really big or really small depending on what gear you are in.

13. Mani_Jha

Sorry, I dont understand what you are saying. That is advanced Physics, I guess

14. anonymous

Not entirely. Gear ratios are taught early on. Gear ratio is defined as${\omega_A \over \omega_B} = {r_B \over r_A}$where A is the input side and B is the output speed. For example, if a gear is spinning at 10 rad/s and has a radius of 10 m is connected to another gear of radius 5, we can find the angular velocity of the output gear as$\omega_B = \omega_A *{r_A \over r_B} = 10*2 = 20$ Therefore for a car. Speed of car = engine speed * gear ratio of transmission * gear ratio of differential * 2 pi * radius of tire.

15. Mani_Jha

Ok, I get gear ratio, but what is gear ratio in transmission and gear ratio in differential? Sorry, If I m troubling you

16. anonymous

No problem. I shouldn't have gotten off-topic earlier. The gear ratios will be different depending on the car. You can usually find them online if you want to run the speedometer/tachometer relationship experiment you described earlier. Modern overdrive transmissions typically have gear ratios ranging from 5:1 in first gear to 0.75:1 in the overdrive gear (4,5,or even 8th gear in modern auto-transmissions). The differential is the device that splits the torque between to two drive tires. Typical differential gear ratios for light-duty, rear wheel drive trucks is 3.77. This number will be higher for fuel-efficient, small engine, front wheel drive cars, like a Honda Accord. Typically, car manufacture websites list the gear ratios in the transmission. The "final drive ratio" is the differential gear ratio. To find the correlation between the vehicles speed and engine speed, we can use the following relationship. $v_c = {2 \pi r_t \times RPM * 2 \pi/60 \over gr_t \times gr_d}$where $$r_t$$ is the radius of your tires in meters, $$RPM$$ is the engine speed in revolutions per minute, $$gr_t$$ is the gear ratio of the current gear of the transmission, and $$gr_d$$ is the gear ratio of the differential. $$v_c$$ will be in units of m/s. Convert to other velocity units if desired.

17. Mani_Jha

I dont know how to thank you, but still, thanks a lot!

18. anonymous

Not a problem. I've actually performed the experiment you described. I knew the tire size and transmission gear ratio. I calculated the differential gear ratio from the vehicle speed and engine speed I noted from my dashboard.