a spring of force constant 'k' is cut into 2 parts whose lengths are in the ratio 1:2.The 2 parts are now connected in parallel and a block of mass 'm' is suspended at the end of the combined spring.find the period of oscillation performed by the block
Spring constant is inversely proportional to spring length. The original spring constant k becomes two springs with constants k1 = 3k and k2 = 3k/2, in parallel.
The effective spring constant of springs in parallel is obtaned by adding the separate spring constants; so the new spring constant becomes
k' = k1 + k2 = 9k/2.
The new period of oscillation is:
P = 2*pi*sqrt(m/k')
= 2*pi*sqrt(2/9)*sqrt(m/k)
= (2/3)*pi*sqrt(2m/k)
how is k1=3k and k2=3k/2.i am getting it as k/3 and 2k/3
To find the period of oscillation performed by the block, we can follow these steps:
1. Determine the effective force constant of the combined spring:
The force constant of a spring is given by the equation F = kx, where F is the force applied to the spring and x is the displacement of the spring from its equilibrium position. Since we have cut the original spring into two parts, the individual force constants of the two parts will change. Let's denote the force constants of the two parts as k₁ and k₂.
Given that the lengths of the two parts are in the ratio 1:2, we can assume that the force constants also follow the same ratio. Therefore, k₁:k₂ = 1:2.
When the two parts are connected in parallel, the effective force constant of the combined spring, k_eff, can be calculated using the following formula:
1/k_eff = 1/k₁ + 1/k₂
=> 1/k_eff = 1/k₁ + 1/(2k₁)
=> 1/k_eff = 3/(2k₁)
=> k_eff = 2k₁/3
2. Determine the effective mass of the system:
Since we are considering a block of mass 'm' suspended at the end of the combined spring, the effective mass, m_eff, will be equal to the actual mass 'm'. The mass does not change when the spring is cut.
3. Find the period of oscillation:
The period of oscillation for a mass-spring system can be calculated using the formula:
T = 2π√(m_eff/k_eff)
Plugging in the values we obtained earlier, we get:
T = 2π√(m/(2k₁/3))
=> T = 2π√(3m/2k₁)
Therefore, the period of oscillation performed by the block is 2π√(3m/2k₁).