0.460 kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to change the period to 1.95 s
0.460 [1.95 / 1.50)^2 - 1] = ?
To calculate the additional mass needed to change the period, we can use the formula to determine the period of a mass-spring system:
T = 2π√(m/k)
Where:
T = Period
m = Mass
k = Spring constant
Given:
T1 = 1.50 s
T2 = 1.95 s
m1 = 0.460 kg
Let's start by rearranging the formula to solve for the spring constant:
k = (4π²m) / T²
Now, we can find the spring constant using the initial period:
k1 = (4π² * 0.460 kg) / (1.50 s)²
Next, we can determine the additional mass needed to achieve the desired period:
T2 = 2π√((m1 + m2) / k1)
Rearranging the formula, we get:
(m1 + m2) = (T2 / 2π)² * k1
Now, let's substitute the given values into the equation:
(m1 + m2) = (1.95 s / 2π)² * (4π² * 0.460 kg) / (1.50 s)²
Simplifying the equation:
(m1 + m2) = (1.95 s / 2π)² * (4π² * 0.460 kg) / (2.25 s²)
(m1 + m2) = (1.19)² * (4 * 0.460 kg) / 2.25
(m1 + m2) = 1.414 * 0.92 kg
(m1 + m2) ≈ 1.30 kg
Therefore, approximately 1.30 kg of additional mass must be added to change the period to 1.95 s.
To solve this problem, we can use the relationship between the period and the mass for a mass-spring system, which is given by:
T = 2π√(m/k)
where T is the period, m is the mass, and k is the spring constant.
Given that the initial period (T1) is 1.50 s and the final period (T2) is 1.95 s, we can set up the equation as follows:
T1 = 2π√(m1/k)
T2 = 2π√(m2/k)
To find the mass that must be added to change the period, we need to find the difference in the mass between the two scenarios. This can be done by subtracting the first equation from the second equation:
T2 - T1 = 2π√(m2/k) - 2π√(m1/k)
Next, we can isolate the difference in mass (Δm) by dividing both sides of the equation by 2π:
(T2 - T1) / (2π) = √(m2/k) - √(m1/k)
Now we square both sides of the equation to eliminate the square root:
[(T2 - T1) / (2π)]^2 = (m2/k) - 2√(m2/k)√(m1/k) + (m1/k)
Simplifying further,
[(T2 - T1) / (2π)]^2 = (m2 + m1)/k - 2√(m2m1)/k
Now, let's solve this equation for Δm (the difference in mass):
Δm = k * [(T2 - T1) / (2π)]^2 - m1
To obtain the mass that must be added to change the period, substitute the known values into the equation and solve for Δm.