A block hangs in equilibrium from a vertical spring. When a second identical block is added, the original block sags by 5.00 cm. What is the oscillation frequency of the two block system?
To find the oscillation frequency of the two-block system, we need to use Hooke's Law and the equation for the oscillation frequency of a mass-spring system.
Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position. Mathematically, it can be expressed as:
F = -kx
Where:
- F is the force applied by the spring,
- k is the spring constant,
- x is the displacement from the equilibrium position.
In this case, when a single block hangs in equilibrium from the spring, the force exerted by the spring is balanced by the weight of the block. So, we have:
mg = kx1
Where:
- m is the mass of each block,
- g is the acceleration due to gravity,
- x1 is the equilibrium displacement of the spring.
When a second identical block is added, the equilibrium displacement of the spring increases to x2 = x1 + 0.05 m.
Now, we need to find the new force exerted by the spring. Since two blocks are hanging now, the force exerted by the spring is:
2mg = kx2
We can substitute x2 = x1 + 0.05 m into the equation:
2mg = k(x1 + 0.05)
Now, we can solve for k:
k = (2mg) / (x1 + 0.05)
Next, we can calculate the oscillation frequency of the mass-spring system using the equation:
f = (1 / 2π) * sqrt(k / m)
Where:
- f is the oscillation frequency,
- π is a mathematical constant (approximately 3.14159).
Plugging in the values of k and m, we get:
f = (1 / 2π) * sqrt((2mg) / (x1 + 0.05) / m)
Simplifying the equation further, we have:
f = (1 / 2π) * sqrt((2g) / (x1 + 0.05))
Finally, substitute the given value of x1 (0.05 m) into the equation, and calculate the oscillation frequency using the known value of g (acceleration due to gravity, approximately 9.8 m/s^2).
F=kx = mg
x=mg/k
but for the second block we have
x+.05=2mg/k
setting the x equal..
mg/k-2mg/k=-.05
-mg/k=-.05
k= mg/.05
now frequency:
f= 1/2pi sqrt (k/m)
put that k int the expression, and you have the answer.