Concepts Of Physics MCQ Edition [Volume 1]PhysicsSimple Harmonic Motion
Determine the elastic potential energy stored in each spring shown in the figure when the block is in equilibrium. Additionally, determine the time period of vertical oscillation of the block.
Options
- AEnergies: M^2 g^2 k₁ , M^2 g^2 k₂ and M^2 g^2 k₃ ; Time period: 2 M ( 1 k₁ + 1 k₂ + 1 k₃ )
- BEnergies: M g^2 2 k₁ , M g^2 2 k₂ and M g^2 2 k₃ ; Time period: 2 M (k₁ + k₂ + k₃)
- CEnergies: M^2 g^2 2 k₁ , M^2 g^2 2 k₂ and M^2 g^2 2 k₃ ; Time period: 2 M k₁ + k₂ + k₃
- DEnergies: M^2 g^2 2 k₁ , M^2 g^2 2 k₂ and M^2 g^2 2 k₃ ; Time period: 2 M ( 1 k₁ + 1 k₂ + 1 k₃ )
Correct answer
D. Energies: M^2 g^2 2 k₁ , M^2 g^2 2 k₂ and M^2 g^2 2 k₃ ; Time period: 2 M ( 1 k₁ + 1 k₂ + 1 k₃ )
Step-by-step solution
In equilibrium, the tension in each spring is equal to the weight of the block, Mg . The extension in each spring is given by: x₁ = Mg k₁ , x₂ = Mg k₂ , x₃ = Mg k₃ The elastic potential energy stored in each spring is U = 1 2 k x^2 . U₁ = 1 2 k₁ ( Mg k₁ )^2 = M^2 g^2 2 k₁ U₂ = 1 2 k₂ ( Mg k₂ )^2 = M^2 g^2 2 k₂ U₃ = 1 2 k₃ ( Mg k₃ )^2 = M^2 g^2 2 k₃ Since the springs are connected in series, the equivalent spring constant k_ eq is given by: 1 k_ eq = 1 k₁ + 1 k₂ + 1 k₃ The time period of vertical oscillation is: T =