Original Question:
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?
bobpursley s'ltn:
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
Okay bob, I did that and got 0.112 when my calculator was in degrees, but I know it needs to be in radians. So I got 21.99 when my calculator was in radians. I think I may just be entering it into my calculator incorrectly.
When I initially did it I got 2.23Hz.. which was also incorrect.
To calculate the oscillation frequency of the two block system, we first need to determine the spring constant (k). From bobpursley's solution, we have already calculated k = mg/0.05.
To convert the frequency from degrees to radians, you need to make sure your calculator is in the correct mode. Most calculators default to radians, but if yours is in degrees, you'll need to switch it to radians mode.
Once your calculator is in radians mode, you can use the formula:
f = 1 / (2 * π) * √(k/m)
Substitute the value of k that you calculated (mg/0.05) and the mass of the blocks (m) into the formula to determine the oscillation frequency.
To calculate the oscillation frequency of the two-block system, you'll need to follow a series of steps:
Step 1: Calculate the spring constant (k):
First, use the given information to find the value of k. The formula F = kx represents Hooke's Law, where F is the force applied by the spring, k is the spring constant, and x is the displacement from the equilibrium position. In this case, the force applied by the spring is equal to the weight of the block(s), so F = mg.
For the single block in equilibrium, we can write:
kx = mg
Rearranging the equation:
x = mg/k
Step 2: Determine the displacement for two blocks:
When a second identical block is added, the original block sags by 5.00 cm. So the displacement of the second block (x2) is equal to the original displacement (x) plus 5.00 cm.
x2 = x + 0.05 m
Step 3: Set up the equation:
Setting the expressions for x equal to each other:
mg/k = 2mg/k - 0.05
Rearranging the equation:
mg/k - 2mg/k = -0.05
Step 4: Solve for the spring constant (k):
Simplifying the equation:
-mg/k = -0.05
Dividing both sides by -1:
mg/k = 0.05
Isolating k:
k = mg/0.05
Step 5: Calculate the oscillation frequency (f):
Now that you have the value of k, plug it into the formula for the oscillation frequency:
f = 1 / (2π * sqrt(k/m))
Substituting the value of k and simplifying the expression will give you the final answer.
Regarding your calculator, make sure it is set to radians mode when performing trigonometric calculations. If you obtained an answer of 0.112 when your calculator was in degrees, converting it to radians would be incorrect. Double-check your calculator settings and try entering the expression again to get the correct value for the oscillation frequency.