When a mass of 21 g is attached to a certain
spring, it makes 16 complete vibrations in
4.4 s.
What is the spring constant of the spring?
Answer in units of N/m
The period is
P = 2*pi*sqrt(m/k) = 4.4/16 = 0.0275 s
The mass is
m = 0.021 kg
Solve for k by rearranging the first formula
To find the spring constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, this can be expressed as:
F = -kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.
In this case, we are given that the mass attached to the spring makes 16 complete vibrations in 4.4 seconds. Each complete vibration corresponds to one period, T, of oscillation. The period can be calculated by dividing the total time for 16 vibrations by the number of vibrations:
T = (4.4 s) / 16 = 0.275 s
The period, T, of an oscillating mass-spring system is related to the spring constant, k, and the mass, m, by the equation:
T = 2π(√(m/k))
Rearranging the equation to solve for the spring constant, k, we have:
k = (4π²m) / T²
Plugging in the values, we have:
k = (4π² * 0.021 kg) / (0.275 s)²
Calculating this expression, we find:
k ≈ 75.6 N/m
Therefore, the spring constant of the spring is approximately 75.6 N/m.
To find the spring constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. Mathematically, this law can be expressed as:
F = -kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, we have information about the mass attached to the spring and the number of vibrations in a given time period. We can use these values to calculate the spring constant.
1. First, let's determine the time period (T) for one complete vibration:
The total time for 16 complete vibrations is 4.4 s.
So, the time period for one complete vibration (T) can be calculated as:
T = Total time / Number of vibrations
T = 4.4 s / 16
T ≈ 0.275 s
2. The time period (T) is related to the frequency (f) by the formula:
f = 1 / T
f = 1 / 0.275 s
f ≈ 3.64 Hz
(Note: Hz represents the unit of frequency, which is a reciprocal second)
3. The angular frequency (ω) can be calculated from the frequency using the formula:
ω = 2πf
ω = 2π × 3.64 Hz
ω ≈ 22.92 rad/s
(Note: rad/s represents the unit of angular frequency)
4. The spring constant (k) can be calculated using the formula:
k = mω²
where m is the mass attached to the spring.
Given mass = 21 g = 0.021 kg (converting grams to kilograms)
k = 0.021 kg × (22.92 rad/s)²
k ≈ 11.8 N/m
Therefore, the spring constant of the spring is approximately 11.8 N/m.