An aircraft piston engine starts from rest and accelerates to 1500rpm in 4.95s. This engine through a reduction gearbox drives a 9m diameter propeller at 55rpm

What is the engine's angular displacement in 4.95s?

To find the engine's angular displacement in 4.95 seconds, we need to know the engine's initial and final angular velocities.

Given:
Initial angular velocity (ω₁) = 0 rpm (since the engine starts from rest)
Final angular velocity (ω₂) = 1500 rpm
Time (t) = 4.95 seconds

Angular displacement (θ) can be calculated using the formula:

θ = ω₁t + 0.5 α t²

Here, α represents the angular acceleration. Since we don't have the angular acceleration, we need to find it first using the given information.

First, let's convert the initial and final angular velocities to radians per second (rad/s), as we need to work with SI units:

Initial angular velocity (ω₁) = (0 rpm) * (2π rad/1 min) * (1 min/60 s) = 0 rad/s
Final angular velocity (ω₂) = (1500 rpm) * (2π rad/1 min) * (1 min/60 s) = 157.08 rad/s

Now, let's calculate the angular acceleration:

α = (ω₂ - ω₁) / t
= (157.08 rad/s - 0 rad/s) / 4.95 s
= 31.7 rad/s² (rounded to one decimal place)

Using the calculated angular acceleration, we can find the angular displacement:

θ = ω₁t + 0.5 α t²
= (0 rad/s) * (4.95 s) + 0.5 * (31.7 rad/s²) * (4.95 s)²
= 0.5 * (31.7 rad/s²) * (4.95 s)²
≈ 387 rad (rounded to three significant figures)

Therefore, the engine's angular displacement in 4.95 seconds is approximately 387 radians.

The angular displacement can be found by multiplying the angular velocity by the time. The angular velocity is the change in angular displacement per unit of time.

In this case, the engine accelerates from rest to 1500 revolutions per minute (rpm) in 4.95 seconds. To get the angular velocity, we need to convert 1500 rpm to radians per second (rad/s). Since there are 2π radians in one revolution, we can use the following conversion:

Angular velocity = (1500 rpm × 2π rad/1 min) / 60 s

Let's calculate the angular velocity:

Angular velocity = (1500 rpm × 2π rad/1 min) / 60 s
Angular velocity = 1500 × 2π / 60 rad/s

Now, to find the angular displacement, we multiply the angular velocity by the time:

Angular displacement = Angular velocity × Time
Angular displacement = (1500 × 2π / 60) rad/s × 4.95 s

Let's calculate the angular displacement:

Angular displacement = (1500 × 2π / 60) rad/s × 4.95 s
Angular displacement = (1500 × 2π × 4.95) / 60 rad
Angular displacement ≈ 157.08 rad