What is the natural frequency of the spring mass system shown below? The contact between the block and the inclined plane is frictionless. The mass of the block is denoted by m and the spring constants are denoted by k₁ and k₂ as shown below.

Question

Question

\(\sqrt {\frac{{{k_1} + {k_2}}}{{2m}}}\)
\(\sqrt {\frac{{{k_1} + {k_2}}}{{4m}}}\)
\(\sqrt {\frac{{{k_1} - {k_2}}}{m}}\)
\(\sqrt {\frac{{{k_1} + {k_2}}}{2m}}\)
\(\sqrt {\frac{{{k_1} + {k_2}}}{m}}\)

Answer 1

Concept:

Let the mass is displaced x downwards in the direction of the wedge.

Spring force F₁ = k₁x and F₂ = k₂x acts in the direction opposite to the direction of displacement force.

Net force, F = (k1 + k2)x

As the system is in equilibrium

mẍ + F = 0

⇒ mẍ + (K₁ + K₂)x = 0

⇒ mẍ \( + \left( {\frac{{{k_1} + {k_2}}}{m}} \right)x = 0\)

\(\therefore {\omega _n} = \sqrt {\frac{{{k_1} + {k_2}}}{m}} \)