Chapter 8, Problem 40P

(a)

To determine

To find: the moment of inertia of the three rotor blades about the axis of rotation

Answer to Problem 40P

The value is $2250\text{kg}\cdot {\text{m}}^{\text{2}}$ .

Explanation of Solution

Given information:

A helicopter rotor blade can be considered a long thin rod, If each of the three rotor helicopter blades is $3.75\text{m}$ long and has a mass of $160\text{kg}$ ,

Formula used:

  $I=\frac{1}{3}M{L}^2+\frac{1}{3}M{L}^2+\frac{1}{3}M{L}^2$

Here,

  $I$ is the moment of inertia,

  $M$ is the mass of the rod,

  $L$ is the length of rod.

Calculation:

Length of each rotor blade of helicopter $L=3.75\text{m}$

Mass of each blade $M=160\text{kg}$

The moment of inertia, l of the rotor about axis of rotation

  $\begin{array}{l}I=\frac{1}{3}M{L}^2+\frac{1}{3}M{L}^2+\frac{1}{3}M{L}^2\\ I=M{L}^2\\ =160\times {\left(3.75\right)}^2\text{kg}\cdot {\text{m}}^{\text{2}}\\ I=2250\text{kg}\cdot {\text{m}}^{\text{2}}\end{array}$

Conclusion:

The value is $2250\text{kg}\cdot {\text{m}}^{\text{2}}$ .

(b)

To determine

To find: The value of the torque.

Answer to Problem 40P

The value is $\tau =8.8\times {10}^3\text{N}\cdot \text{m}$ .

Explanation of Solution

Given information:

A helicopter rotor blade can be considered a long thin rod, as shown If each of the three rotor helicopter blades is $3.75\text{m}$ long and has a mass of $160\text{kg}$ ,

Formula used:

  $\tau =I\alpha$

Here,

  $I$ is the moment of inertia,

  $\tau$ is the torque,

  $\alpha$ is the angular acceleration.

Calculation:

Speed of the blades $\omega =5\text{rev/s}$ in $8.0\text{s}$

  $=10\pi \text{rad/s}$

  $\begin{array}{l}\alpha =\frac{\Delta \omega }{\Delta t}\\ \alpha =\frac{10\pi }{8}\\ \text{or}=\frac{5\pi }{4}{\text{rad/s}}^{\text{2}}\end{array}$

Hence, the torque to be applied to attain the desired speed

  $\begin{array}{l}\tau =I\alpha \\ =\left(2250\right)\left(\frac{5\pi }{4}\right)\text{N}\cdot \text{m}\\ \tau =8.8\times {10}^3\text{N}\cdot \text{m}\end{array}$

Conclusion:

The value is $\tau =8.8\times {10}^3\text{N}\cdot \text{m}$ .