To determine the angular velocity of a solid body that starts rotating about a stationary axis with a given angular acceleration, we can use the relationship between angular acceleration, angular velocity, and the angle of rotation. In this case, the angular acceleration is defined as α = 0 cos(θ), where 0 is a constant vector and θ is the angle of rotation from the initial position. Let's break this down step by step.

Understanding Angular Acceleration

Angular acceleration (α) is the rate of change of angular velocity (ω) with respect to time. In our scenario, we have:

• α = dω/dt
• α = 0 cos(θ)

This means that the angular acceleration depends on the angle θ. As the body rotates, the angular acceleration changes based on the cosine of the angle.

Relating Angular Velocity and Angular Acceleration

To find the angular velocity as a function of the angle, we can use the relationship between angular acceleration and angular velocity. We know that:

• α = dω/dt = dω/dθ * dθ/dt = dω/dθ * ω

From this, we can rearrange the equation to express it in terms of ω:

• dω = α(1/ω) dθ

Substituting Angular Acceleration

Now, substituting α = 0 cos(θ) into the equation gives us:

• dω = 0 cos(θ) (1/ω) dθ

This can be rearranged to:

• ω dω = 0 cos(θ) dθ

Integrating to Find Angular Velocity

Next, we integrate both sides. The left side integrates with respect to ω, and the right side integrates with respect to θ:

• ∫ ω dω = ∫ 0 cos(θ) dθ

Integrating the left side gives:

• (1/2)ω² = 0 sin(θ) + C

Here, C is the constant of integration. To find C, we can consider the initial conditions. If the body starts from rest, when θ = 0, ω = 0, thus:

• 0 = 0 sin(0) + C → C = 0

Final Expression for Angular Velocity

Now we can express the relationship without the constant:

• (1/2)ω² = 0 sin(θ)

Multiplying both sides by 2 gives:

• ω² = 2 * 0 sin(θ)

Taking the square root to find ω, we have:

• ω = √(2 * 0 sin(θ))

Summary of Results

In conclusion, the angular velocity of the solid body as a function of the angle θ is given by:

• ω(θ) = √(2 * 0 sin(θ))

This equation shows how the angular velocity increases as the angle of rotation increases, influenced by the sine function, which reflects the nature of the angular acceleration provided in the problem. This relationship is crucial in understanding rotational dynamics and can be applied to various physical systems where angular motion is involved.