A mass m is gently placed on the end of a freely hanging spring. The mass then falls 29.8 cm before it stops and begins to rise. What is the frequency of the oscillation?
To find the frequency of the oscillation, we can use the formula:
f = (1 / 2π) * √(k / m)
where:
- f is the frequency of oscillation,
- k is the spring constant, and
- m is the mass.
To find the spring constant, we need to calculate the force exerted by the spring when the mass is stretched.
The force exerted by the spring is given by Hooke's Law:
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, the mass falls 29.8 cm before it stops and starts to rise. This means the total displacement is twice that distance, or 2 * 29.8 cm.
So, x = 2 * 29.8 cm = 59.6 cm = 0.596 m.
The force exerted by the spring is equal to the weight of the mass:
F = mg
where:
- m is the mass, and
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the mass is m, the force F is equal to mg.
Substituting this into the equation F = -kx, we have:
mg = -kx
Solving for k gives:
k = -mg / x
Now we can substitute the values into the equation for the frequency:
f = (1 / 2π) * √(k / m)
f = (1 / 2π) * √((-mg / x) / m)
f = (1 / 2π) * √(-g / x)
Substituting the values for g and x:
f = (1 / 2π) * √(-9.8 m/s^2 / 0.596 m)
Evaluating the expression gives:
f ≈ 1.665 Hz
Therefore, the frequency of the oscillation is approximately 1.665 Hz.
To find the frequency of oscillation, we need to use Hooke's Law and the equation for the period of a mass-spring system. Here's how you can calculate it:
1. Start by finding the spring constant (k) of the spring using Hooke's Law.
- 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, k is the spring constant, and x is the displacement.
- In this case, we can assume that when the mass is at its equilibrium position, the spring is neither stretched nor compressed. Therefore, the displacement (x) will be the distance the mass falls before stopping, which is 29.8 cm.
- Since the force in the downward direction is equal to the weight of the mass (mg), we can write mg = -kx.
- Isolating k, we get k = -mg / x.
2. Calculate the spring constant (k) using the given values.
- Substitute the mass (m) and the displacement (x) into the equation k = -mg / x.
- Make sure to convert the displacement to meters (divide by 100) and the mass to kilograms (divide by 1000) to maintain consistent units.
3. Determine the period (T), which is the time taken for one complete oscillation.
- The period can be calculated using the equation T = 2π√(m/k), where π is approximately 3.14159, m is the mass, and k is the spring constant.
- Plug in the values for the mass (m) and the spring constant (k) to calculate the period.
4. Finally, find the frequency (f) by taking the reciprocal of the period.
- The frequency of oscillation (f) is given by the equation f = 1/T, where T is the period.
- Calculate the reciprocal of the period to get the frequency.
By following these steps, you should be able to find the frequency of the oscillation.
f= 1/2PI * sqrt (k/m)
but k= mg/.298
f= 1/2PI * sqrt (g/.298)