Physics Problem:
A lever system has two different configurations:
– The left lever has a mass of 0.5 kg placed 2 meters from the fulcrum.
– The right lever has a mass of 1 kg placed 1 meter from the fulcrum.
Both levers experience a pulling force of 1 N applied at the same distance from the fulcrum on the opposite side.
Q1: Which of the two levers is harder to pull?
Q2: Derive the relationship between the velocities of the two masses using equations of motion.
Detailed Solution:
Step 1: Compare Moments of Inertia
Moment of inertia determines how much torque is needed to rotate the system. It depends on both the mass and the distance from the fulcrum.
I = m × r²
For the left lever:
I₁ = 0.5 × (2)² = 2 kg·m²
For the right lever:
I₂ = 1 × (1)² = 1 kg·m²
Conclusion: Since I₁ > I₂, the left lever is harder to pull.
Step 2: Use Newton’s Second Law to Find Acceleration
F = ma ⇒ a = F/m
For the left lever:
a₁ = 1 / 0.5 = 2 m/s²
For the right lever:
a₂ = 1 / 1 = 1 m/s²
Step 3: Use Kinematic Equation to Find Velocity
Assuming initial velocity u = 0 and displacement s:
v² = 2as
Left lever:
v₁ = √(2 × 2 × s) = √(4s) = 2√s
Right lever:
v₂ = √(2 × 1 × s) = √(2s)
Velocity Ratio:
v₁ / v₂ = 2√s / √2s = √2
Conclusion: The velocity of the lighter mass on the left lever is √2 times that of the heavier mass on the right.
Final Answers:
- Q1: The left lever is harder to pull because it has a greater moment of inertia.
- Q2: The velocity of the lighter mass is √2 times the velocity of the heavier mass: v₁ = √2 × v₂