Chapter 4.1, Problem 56E

Solution

a.

To find: The angular speed of the propeller blade in radians per second.

The angular speed of the propeller blade in radians per second is $4.5\pi$ .

Given information:

The propeller turns at a speed of $135\text{ rpm}$ .

Radius $R=1.2\text{ m}$ .

Calculation:

The angular speed in revolutions per minute is converted into revolutions per second.

  $\left(\frac{135\text{revolutions}}{1\text{minute}}\right)\left(\frac{1\text{ minute}}{60\text{ seconds}}\right)=2.25\text{revolutions}/\text{second}$

Converting revolutions per second into radians per second,

  $\left(\frac{2.25\text{ revolutions}}{1\text{ second}}\right)\left(\frac{2\pi \text{ radians}}{1\text{ revolution}}\right)=4.5\text{radians}/\text{second}$

Therefore, the angular speed of the propeller blade in radians per second is $4.5\pi$ .

b.

To find: The linear speed of the propeller blade in meters per second.

The linear speed of the propeller blade in meters per second is $16.84$ .

Given information:

The propeller turns at a speed of $135\text{ rpm}$ .

Radius $R=1.2\text{ m}$ .

Calculation:

Let the velocity $v$ is given by,

  $v=\frac{\Delta s}{\Delta t}$

Where,

  $\Delta s$ is the displacement along the circumference

  $\Delta t$ is the time taken for the revolution that occurs

It is given that $135$ revolutions occur in one minute.

Thus,

  $\begin{array}{c}v=\frac{\Delta s}{\Delta t}\\ =\frac{135\times 2.4\pi \text{ meters}}{1\text{ minute}}\\ =\left(\frac{135\times 2.4\pi \text{ meters}}{1\text{ minute}}\right)\left(\frac{1\text{ minute}}{60\text{ seconds}}\right)\\ v\approx 16.84\text{meters}/\text{second}\end{array}$

Therefore, the linear speed of the propeller blade in meters per second is $16.84$ .

c.

To find: The linear speed of a point on blade halfway between the center of the propeller and the tip of the blade.

The linear speed of a point on blade halfway between the center of the propeller and the tip of the blade in meters per second is $8.43$ .

Given information:

The propeller turns at a speed of $135\text{ rpm}$ .

Radius $R=1.2\text{m}$ .

Calculation:

Let the velocity $v$ is given by,

  $v=\frac{\Delta s}{\Delta t}$

Where,

  $\Delta s$ is the displacement along the circumference

  $\Delta t$ is the time taken for the revolution that occurs

Taking half the result from part (b),

Thus,

  $\begin{array}{c}v=\frac{\Delta s}{\Delta t}\\ =\frac{135\times 1.2\pi \text{ meters}}{1\text{ minute}}\\ v\approx 8.43\text{meters}/\text{second}\end{array}$

Therefore, linear speed of a point on blade halfway between the center of the propeller and the tip of the blade in meters per second is $8.43$ .