An accelerometer in a control system consists of a 4.70 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 5.30 m/s2, the object must be located 0.220 cm from its equilibrium position. Find the force constant required for the spring.
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 of the object 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 of the object from its equilibrium position.
In this case, the acceleration of the sliding object is caused by the spring. The force exerted by the spring is equal to the mass of the object multiplied by its acceleration:
F = m * a
Where:
F is the force exerted by the spring,
m is the mass of the object,
a is the acceleration.
The mass of the object is given as 4.70 g. Since the force constant k is given in N/m, we need to convert the mass to kg:
m = 4.70 g = 0.00470 kg
The acceleration is given as 5.30 m/s^2.
Now we can equate the force exerted by the spring (F = m * a) to Hooke's Law (F = -kx) to find the force constant:
m * a = -k * x
Rearranging the equation, we get:
k = -m * a / x
Substituting the given values, we have:
k = -(0.00470 kg) * (5.30 m/s^2) / (0.00220 m)
Calculating this expression, we find:
k ≈ -48 N/m
Therefore, the force constant required for the spring is approximately 48 N/m.