Final answer:
To calculate the magnitudes of the initial accelerations of particles 1 and 2, we can use the equation for rotational motion and the concept of moment of inertia.
Step-by-step explanation:
When the rod is released, it will rotate about the fulcrum due to the difference in distances of the masses from the fulcrum. The initial accelerations of particles 1 and 2 can be calculated using the equation for rotational motion:
α = (mass of particle × distance from fulcrum) / (moment of inertia)
Since the rod is rigid and massless, the moment of inertia can be calculated as:
Moment of inertia = (m₁ × L₁² + m₂ × L₂²)
Substituting the expressions for torque and moment of inertia into the torque equation:
m × g × L₁ + m × g × L2 = (m × L₁ ² + m × L₂²) × α
Solving for α:
α = g × (L₁ + L₂) / (L₁² + L₂²)
Calculate the initial accelerations:
The initial accelerations of the particles are related to the angular acceleration by:
a₁ = α × L₁
a₂ = α × L₂
7. Plug in the values and calculate the initial accelerations:
α = (9.81 m/s²) × (0.2 m + 0.8 m) / (0.2² m² + 0.8² m²) ≈ 4.76 rad/s²
a₁ = 4.76 rad/s² × 0.2 m ≈ 0.95 m/s²
a₂ = 4.76 rad/s² × 0.8 m ≈ 3.81 m/s²
Therefore, the initial accelerations of particles 1 and 2 are approximately 0.95 m/s² and 3.81 m/s², respectively.