The blades of an aeroplane propeller are 2m long and rotate at 300 rpm. Calculate (i) frequency (ii) period of rotation (iii) angular velocity (iv) linear velocity of a point 0.5 m from the top of the blades.

Answer

Let's solve the problem step by step.

### Given Data:
- Length of the propeller blade (L) = 2 m
- Rotations per minute (RPM) = 300

### Step 1: Calculate Frequency (f)
Frequency is defined as the number of rotations per second.

1. Convert RPM to rotations per second:
\[ \text{Rotations per second} = \frac{300 \text{ rotations}}{60 \text{ seconds}} = 5 \text{ rotations/second} \]
Therefore, the frequency \( f \) is:
\[ f = 5 \text{ Hz} \]

### Step 2: Calculate Period of Rotation (T)
The period of rotation is the reciprocal of frequency.

1. Use the formula:
\[ T = \frac{1}{f} = \frac{1}{5} = 0.2 \text{ seconds} \]

### Step 3: Calculate Angular Velocity (ω)
Angular velocity can be calculated using the formula:
\[ \omega = 2\pi f \]

1. Substitute the value of frequency:
\[ \omega = 2\pi \times 5 = 10\pi \text{ radians/second} \]

### Step 4: Calculate Linear Velocity (v) at a Point 0.5 m from the Top of the Blades
To find the linear velocity at a point 0.5 m from the top of the blade, we first need to determine the distance from the axis of rotation.

1. The distance from the axis of rotation is:
\[ \text{Distance from the axis} = \text{Total length of blade} - \text{Distance from the top} = 2 \text{ m} - 0.5 \text{ m} = 1.5 \text{ m} \]

2. Use the formula for linear velocity:
\[ v = \omega \times r \]
where \( r \) is the distance from the axis of rotation.

3. Substitute the values:
\[ v = 10\pi \times 1.5 = 15\pi \text{ meters/second} \]

### Final Answers:
1. Frequency \( f = 5 \text{ Hz} \)
2. Period of rotation \( T = 0.2 \text{ seconds} \)
3. Angular velocity \( \omega = 10\pi \text{ radians/second} \)
4. Linear velocity \( v = 15\pi \text{ meters/second} \)