The object weighing 20 N is hung on the bottom of the spiral spring , so that stretch as far as 9.8 cm after a state of balance , the weight pulled down as far as 5 cm , measured from idle after burdened . Find the equation of motion generated assuming no friction and air resistance
k = 20 N/.098 m = 204 N/m
m = 20/9.81 = 2.04 kg
omega = w = sqrt (k/m) radians/second
so
w = sqrt (204/2.04) = 10 rad/s
if x is distance from equilibrium position, positive up
x = -.05 cos 10 t
To find the equation of motion for the object hanging on the spring, we can apply Hooke's Law and Newton's Second Law.
Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be written 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, the weight of the object hanging on the spring provides the force pulling it down. The weight can be calculated as:
W = mg
where W is the weight, m is the mass of the object, and g is the acceleration due to gravity. In this equation, we use the value of weight as 20 N.
Since the weight pulled down the spring by 5 cm, the displacement from the equilibrium position is -5 cm (negative sign indicates downward direction). Similarly, when the spring stretches to 9.8 cm, the displacement is -9.8 cm.
To find the spring constant (k), we can use Hooke's Law with either of the displacements:
F = -kx
For the displacement of -5 cm, we have:
20 N = -k(-5 cm)
Simplifying, we get:
20 N = 5k cm
Dividing both sides by 5, we find:
k = 4 N/cm
Now, we can use this value of k to find the equation of motion for the object on the spring.
Let's assume the positive direction is upwards and the equilibrium position is at the origin (0 cm).
At any position x (in cm) from the equilibrium position, the net force acting on the object is the sum of the force due to the weight and the force due to the spring:
F_net = -mg + kx
Using Newton's Second Law (F = ma), where a is the acceleration of the object, we have:
-mg + kx = ma
Since acceleration is the second derivative of position with respect to time (a = d²x/dt²), we differentiate the equation with respect to time twice:
d²x/dt² = (-mg + kx)/m
Simplifying, we get the equation of motion:
d²x/dt² = (-g + (k/m)x)
This is the equation of motion for the object hanging on the spiral spring, assuming no friction and air resistance.
Note: The acceleration due to gravity (g) is approximately 9.8 m/s². So, when using this equation, make sure to convert the units of x accordingly (cm to m) to maintain consistency.