With a block of mass m , the frequency of a block-spring system is 1.8Hz. When 50g is added, the frequency drops to 1Hz . Find m and the spring constant.
To solve this problem, we can use the formula for the frequency of a mass-spring system:
f = 1 / (2π) * √(k / m),
where
f = frequency,
k = spring constant, and
m = mass.
Let's denote the original mass of the block as m1 and the mass added as m2. We are given the following information:
f1 = 1.8 Hz (original frequency),
f2 = 1 Hz (frequency after adding 50 g), and
m2 = 50 g.
1. Calculating the original mass (m1):
f1 = 1.8 Hz
1.8 = 1 / (2π) * √(k / m1) [Note: We leave f1 in its exact form for accuracy.]
Rearranging the equation to solve for m1:
1.8 * 2π = √(k / m1)
Squaring both sides of the equation:
(1.8 * 2π)^2 = k / m1
k = m1 * (1.8 * 2π)^2
2. Calculating the new mass:
m = m1 + m2
m = m1 + 50 g
3. Calculating the new spring constant:
Using the new mass value (m1 + 50 g) and the given frequency (1 Hz):
1 = 1 / (2π) * √(k / (m1 + 50 g)) [Note: We leave f2 in its exact form for accuracy.]
Rearranging the equation to solve for k:
1 * 2π = √(k / (m1 + 50 g))
Squaring both sides of the equation:
(2π)^2 = k / (m1 + 50 g)
k = (m1 + 50 g) * (2π)^2
Now we have two equations for k. We can set them equal to each other since they represent the same spring constant:
m1 * (1.8 * 2π)^2 = (m1 + 50 g) * (2π)^2
Simplifying the equation:
1.8^2 * (m1 * (2π)^2) = (m1 + 50 g) * (2π)^2
1.8^2 * m1 = m1 + 50 g
Expanding and rearranging the equation:
3.24 * m1 = m1 + 0.05
2.24 * m1 = 0.05
Simplifying and solving for m1:
m1 = 0.05 / 2.24
m1 ≈ 0.0223 kg
Now that we have found the value for m1, we can substitute it into either equation for k to find the spring constant. Let's use the equation derived in step 1:
k = m1 * (1.8 * 2π)^2
k ≈ 0.0223 kg * (1.8 * 2π)^2
Calculating the value of k:
k ≈ 0.0223 kg * 3.24
k ≈ 0.072 kg/s²
Therefore, the mass of the block (m) is approximately 0.0223 kg, and the spring constant (k) is approximately 0.072 kg/s².
To find the mass (m) and the spring constant (k), we can use the formula for the frequency of a block-spring system:
f = 1 / (2π) * sqrt(k / m)
Given:
Frequency f1 = 1.8 Hz (when mass m)
Frequency f2 = 1 Hz (when mass m + 50g)
Let's first convert the mass increase from grams to kilograms:
increase_in_mass = 50g = 0.05kg
Using these values, we can set up the following equations:
f1 = 1.8 Hz => 1.8 = 1 / (2π) * sqrt(k / m)
f2 = 1 Hz => 1 = 1 / (2π) * sqrt(k / (m + 0.05))
To simplify the equations, let's solve them step by step.
1. Solve for m in terms of k:
1.8 = 1 / (2π) * sqrt(k / m)
Multiply both sides by 2π:
(2π) * 1.8 = sqrt(k / m)
Square both sides to eliminate the square root:
(2π)² * 1.8² = k / m
k = (2π)² * 1.8² * m
2. Substitute the value of k in terms of m into the second equation:
1 = 1 / (2π) * sqrt((2π)² * 1.8² * m / (m + 0.05))
Now we can solve for m.
1 = 1 / (2π) * sqrt((2π)² * 1.8² * m / (m + 0.05))
Multiply both sides by (2π):
2π = sqrt((2π)² * 1.8² * m / (m + 0.05))
Square both sides to eliminate the square root:
(2π)² = (2π)² * 1.8² * m / (m + 0.05)
Simplify and rearrange the equation:
(m + 0.05) = (2π)² * 1.8² * m / (2π)²
(m + 0.05) = 1.8² * m
Expand the square on the right side:
m + 0.05 = 3.24 * m
Rearrange the equation to isolate the variable m:
2.24 * m = 0.05
m = 0.05 / 2.24
m ≈ 0.0223 kg (rounded to 4 decimal places)
Now, substitute the value of m back into the formula for k:
k = (2π)² * 1.8² * m
k = (2π)² * 1.8² * 0.0223
k ≈ 2.01 N/m (rounded to 2 decimal places)
Therefore, the mass (m) is approximately 0.0223 kg, and the spring constant (k) is approximately 2.01 N/m.