Did my textbook make a mistake when writing these units?

[mymodded]

Homework Statement

A motor has a single loop inside a magnetic field with magnitude 0.870T. If the area of the loop is 300 cm^2 , find the maximum angular speed possible for this motor when connected to a source of emf providing 170V

Relevant Equations

$$\Delta V_{ind} = -\frac{d\phi _B}{dt} = \omega BA \sin(\theta ),
\omega = 2\pi f$$

Sorry if there are mistakes regarding the post itself, this is my first time posting.

This an easy problem to solve, but this isn't what I'm looking for, but first of all, you can plug in the values and solve for ##\omega##, and it equals ##\frac{170 V}{0.87 T(0.0300 m^2)}## = 6513 Hz, my question is, is the unit correct here? I thought that it should be measured in rad/s, but the answer guide put in Hz, which is weird, because Hz here means revolutions/second, which is obviously different from rad/s (by a factor of 2 ##\pi##), also, ##\omega## = 2##\pi##f where f is measured in Hz or 1/s (or more specifically, rev/s) and multiplying the number of revolutions by 2##\pi## gives you the number of radians. So did the textbook write it incorrectly?

 

[Steve4Physics]

Homework Statement: A motor has a single loop inside a magnetic field with magnitude 0.870T. If the area of the loop is 300 cm^2 , find the maximum angular speed possible for this motor when connected to a source of emf providing 170V
Relevant Equations: $$\Delta V_{ind} = -\frac{d\phi _B}{dt} = \omega BA \sin(\theta ),
\omega = 2\pi f$$

Sorry if there are mistakes regarding the post itself, this is my first time posting.

This an easy problem to solve, but this isn't what I'm looking for, but first of all, you can plug in the values and solve for ##\omega##, and it equals ##\frac{170 V}{0.87 T(0.0300 m^2)}## = 6513 Hz, my question is, is the unit correct here? I thought that it should be measured in rad/s, but the answer guide put in Hz, which is weird, because Hz here means revolutions/second, which is obviously different from rad/s (by a factor of 2 ##\pi##), also, ##\omega## = 2##\pi##f where f is measured in Hz or 1/s (or more specifically, rev/s) and multiplying the number of revolutions by 2##\pi## gives you the number of radians. So did the textbook write it incorrectly?

Hi @mymodded and welcome to PF.

You are correct - the value should be in rad/s. Occasionally 'official' answers are wrong.

Also, the answer should be rounded to a suitable number of significant figures. I'd say 6500 rad/s is an appropriate answer.

 

[Mister T]

Radians per second and hertz have the same dimension. They are interchangeable. This is a source of great confusion and much has been written about the fact that the radian is a unit but not a dimension.

If you look up the definition of the radian you find that it is dimensionless, it's a ratio of lengths.

As a student you have to get used to the fact that the radian can pop into and out of your calculations.

Take for example the relation ##v=r\omega##. Calculate the value of ##v## if ##r=2\ \mathrm{m}## and ##\omega=3\ \mathrm{rad/s}##. The value of ##v## is ##6\ \mathrm{m/s}##.

Note: Edited on 29 March to fix the mistake I made, discussed in Post #5.

 

[Steve4Physics]

Radians per second and hertz have the same dimension. They are interchangeable.

Just to ensure that the OP is not confused, we need to be clear that, as units, rad/s and Hz, are not freely interchangeable. A conversion factor is needed. 1Hz = ##2\pi## rad/s.

The OP asked:

and it equals = 6513 Hz, my question is, is the unit correct here?

So the answer to the OP's question is: no - the unit is wrong; the correct answer is 6513rad/s (ignoring the inappropriate number of significant figures).

 

[Mister T]

we need to be clear that, as units, rad/s and Hz, are not freely interchangeable.

My mistake. You are correct. Even though they have the same dimension they are not the same unit, and thus are not interchangeable. I need to be more careful.

 

[mymodded]

Hi @mymodded and welcome to PF.

You are correct - the value should be in rad/s. Occasionally 'official' answers are wrong.

Also, the answer should be rounded to a suitable number of significant figures. I'd say 6500 rad/s is an appropriate answer.

Thanks a lot