An accelerometer in a control system consists of a 2.42 g object sliding on a horizontal rail. A low-mass spring is connected between the object and a flange at one end of the rail. Grease on the rail makes static friction negligible, but rapidly damps out vibrations of the sliding object. When subject to a steady acceleration of 3.50 m/s2, the object must be located 0.370 cm from its equilibrium position. Find the force constant required for the spring
ma=kx
k=ma/x =2.42•10⁻³•3.5/0.37•10⁻²=2.3 N/m
To find the force constant required for the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
Hooke's Law can be written as:
F = -kx
Where:
F is the force exerted by the spring,
k is the force constant (also known as the spring constant),
x is the displacement from the equilibrium position.
In this case, we are given that the object is subject to a steady acceleration of 3.50 m/s² and is displaced 0.370 cm from its equilibrium position.
First, let's convert the displacement from centimeters to meters:
x = 0.370 cm = 0.370/100 = 0.0037 m
Now, we can rearrange Hooke's Law to solve for the force constant:
k = -F/x
The force exerted by the system is given by Newton's second law:
F = ma
Where:
m is the mass of the object,
a is the acceleration.
We are given that the acceleration is 3.50 m/s². To find the mass of the object, we can use the formula for acceleration:
a = F/m
m = F/a
Since the object is experiencing an acceleration due to a force equal to its mass times the acceleration, we can substitute m in terms of F:
F = m * a
F = (F/a) * a
Simplifying, we find:
F = F
This shows that the force exerted by the system is equal to itself, confirming that the mass cancels out.
So, we can proceed to find the force constant:
k = -F/x
k = -(m * a)/x
k = -a*x
Substituting the given values:
acceleration (a) = 3.50 m/s²
displacement (x) = 0.0037 m
k = -3.50 * 0.0037
Calculating this expression, we find:
k ≈ -0.01295 N/m
The force constant required for the spring is approximately 0.01295 N/m