a mass is suspended on a vertical spring oscillates with a period of 0.5 sec .When the mass is attached to the spring and allowed to hang at rest .By how much is the spring stretched?
wouldn't that depend on the spring?
2PI/period=sqrt(k/m)
k/m= (2PI/5)^2
force=kx
x=force/k= mg/k= 9.8*(5/2PI)^2
check that.
To determine how much the spring is stretched, we need to use the equation for the period of oscillation of a mass-spring system.
The period of oscillation, T, is given by the formula:
T = 2π√(m/k)
Where:
- T is the period of oscillation,
- π is the mathematical constant pi (approximately 3.14159),
- m is the mass attached to the spring,
- k is the spring constant.
In this case, we are given the period (T) as 0.5 seconds. To find the stretch, we need to find the spring constant (k) and the mass (m).
Since the mass is attached to the spring and allowed to hang at rest, there is no external force acting on the system. This means that the force of gravity acting on the mass (mg) is balanced by the force exerted by the spring (kx), where x is the displacement of the spring from its equilibrium position.
Therefore, we can equate these forces:
mg = kx
We can rearrange this equation to solve for the stretch (x):
x = (mg)/k
To find the stretch, we need to know the values of mass (m), acceleration due to gravity (g), and the spring constant (k). Please provide these values.
To find the amount by which the spring is stretched, we can use Hooke's Law, which states that the force required to stretch or compress a spring is directly proportional to the displacement of the spring from its equilibrium position.
We can use the formula:
F = k * x
Where:
F is the force applied to the spring (weight of the mass)
k is the spring constant
x is the displacement of the spring from its equilibrium position
Since the mass is at rest when hanging, the weight of the mass will be equal to the force applied to the spring. Thus,
Weight of the mass = k * x
However, we are given the period of oscillation, not the spring constant. To derive the relation between the period and the spring constant, we can use the formula for the period of a mass-spring system:
T = 2π * sqrt(m/k)
Where:
T is the period of oscillation
m is the mass of the object attached to the spring
k is the spring constant
Rearranging the formula, we can solve for the spring constant (k):
k = (4π² * m)/(T²)
Substituting the given values, we have:
k = (4π² * m)/(0.5²)
Now, we can substitute the spring constant (k) and the weight of the mass into the equation:
Weight of the mass = k * x
Solving for x, the displacement of the spring:
x = Weight of the mass / k
x = Weight of the mass / ((4π² * m)/(0.5²))
Simplifying the expression:
x = (Weight of the mass * 0.5²) / (4π² * m)