a helical spring fixed by the upper end carries a load of 5kg attached to its lower end .Mass is set into vibration and is measured to make 2 oscillation per seconds .calculate
1.the stiffness of the spring
2.the acceleration at any condition along the spring
First of all Period (T) = time/ number of oscillation
T= 1/2= 0.5s
1. Calculate the stiffness of the spring (k)
But T = 2π√m/k
0.5= 6.284√5/k
0.5/6.284=√5/k
(0.0795)²= (√5/k)²
0.006320= 5/k
K= 5/0.06320
K= 791.1N/m
2. acceleration (a) =Force/mass
a= 50/5= 10m/s²
To calculate the stiffness of the spring, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed.
1. Stiffness of the Spring:
The formula to calculate the stiffness (k) of a spring is:
k = F / x
Where:
- k is the stiffness of the spring
- F is the force exerted by the spring
- x is the displacement or extension/compression of the spring
In this case, we have a load of 5 kg attached to the lower end of the spring. The force exerted by the spring is equal to the weight of the load, which can be calculated using the equation:
F = m * g
Where:
- F is the force
- m is the mass
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
In this case:
m = 5 kg
g = 9.8 m/s^2
F = 5 kg * 9.8 m/s^2 = 49 N
Now, to calculate the displacement (x), we need to understand that the helical spring is undergoing simple harmonic motion (SHM), and it completes 2 oscillations per second. This means that it goes through one complete cycle in 1/2 second (since 1 oscillation = 1/2 cycle).
The period (T) of SHM can be calculated using the formula:
T = 1 / f
Where:
- T is the period
- f is the frequency
In this case:
f = 2 oscillations per second
T = 1 / 2 = 0.5 seconds
To find the displacement (x), we use the equation of SHM:
x = A * sin(2πft)
Where:
- x is the displacement
- A is the amplitude (maximum displacement)
- f is the frequency
- t is the time
Since the equation relates displacement, frequency, and time, we need more information to determine the displacement and, consequently, the stiffness of the spring.
2. Acceleration at Any Condition Along the Spring:
To find the acceleration at any condition along the spring, we need to know the displacement at that particular condition.
The acceleration of an object undergoing SHM can be calculated using the equation:
a = ω^2 * x
Where:
- a is the acceleration
- ω (omega) is the angular frequency
- x is the displacement
The angular frequency can be calculated using the formula:
ω = 2πf
Where:
- ω is the angular frequency
- f is the frequency
Once we have the displacement (x) for a particular condition along the spring, we can calculate the acceleration (a) using the above equation.
Please provide more information about the particular condition along the spring for a more accurate calculation of the acceleration.