## Rotational Speed

Consider a braking system with a 5 m long brake arm has a 200 N force applied to the arm’s end. If the total brake power is 100 kW, what is the rotational speed in rpm?

##
__
__**Hint**

**Hint**

Brake Power:

$$$\dot{W}_b=2\pi \tau \upsilon =2\pi FR\upsilon $$$

where
$$\tau$$
is torque,
$$\upsilon $$
is rotational speed,
$$F$$
is the force at the end of the brake arm, and
$$R$$
is the brake arm’s length.

##
__
__**Hint 2**

**Hint 2**

$$\upsilon $$
is in
$$rev/sec$$
, so a final answer conversion is needed to achieve the rotational speed in rpm.

Brake Power:

$$$\dot{W}_b=2\pi \tau \upsilon =2\pi FR\upsilon $$$

where
$$\tau$$
is torque in
$$N \cdot m$$
,
$$\upsilon $$
is rotational speed in
$$rev/sec$$
,
$$F$$
is the force at the end of the brake arm, and
$$R$$
is the brake arm’s length. Thus,

$$$\upsilon=\frac{\dot{W}_b}{2\pi FR}=\frac{100,000W}{2\pi(200N)(5m)}$$$

$$$=\frac{100,000kg\cdot m^2}{6,280N\cdot m\cdot s^3}=\frac{100,000kg\cdot m^2}{6,280kg\cdot \frac{m}{s^2}\cdot m\cdot s^3}=15.92\:\frac{rev}{sec}$$$

Finally, to convert the rotational speed to rpm:

$$$\upsilon =15.92\:\frac{rev}{sec}\cdot \frac{60\:sec}{1\:min}=955\:rpm$$$

955 rpm