How do you find spring equation?
The equation of motion for a spring-mass system excited by a harmonic force is. M x ¨ + k x = F cos Where M is the mass, K is the spring stiffness, F is the force amplitude and ω is the angular frequency of excitation.
How do you solve a SHM differential equation?
The differential equation for the Simple harmonic motion has the following solutions:
- x = A sin ω t. (This solution when the particle is in its mean position point (O) in figure (a)
- x 0 = A sin ϕ (When the particle is at the position & (not at mean position) in figure (b)
- x = A sin ( ω t + ϕ )
Why do we use second order differential equations?
A second order differential equation can have infinitely many solutions as the arbitrary constants can take any value. We find two linearly independent solutions of a second order differential equation as their combination gives all possible solutions of the equation and finding only one solution does not suffice.
What is differential equation in mathematics?
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
How many ways can you solve a differential equation?
Differential Equations Solutions The solution that contains as many arbitrary constants as the order of the differential equation is called a general solution. The solution free from arbitrary constants is called a particular solution. There exist two methods to find the solution of the differential equation.
How do you find the terminal velocity of a differential equation?
Taking proper sign of air resistance opposing gravity, we have terminal velocity when acceleration vanishes: dvdt=mg−cv2=0→v=vterminal=√mgc. gets included in the coefficient of tanh function for velocity as an asymptotic value.
What is the motion of a spring?
The motion of a mass on a spring can be described as Simple Harmonic Motion (SHM): oscillatory motion that follows Hooke’s Law.
What affects the period of a spring?
The period of a spring-mass system is proportional to the square root of the mass and inversely proportional to the square root of the spring constant.
What is the difference between SHM and oscillation?
Simple harmonic motion is a special case of oscillations. A simple harmonic motion is possible only in theory, but oscillations are possible in any situation. The total energy of the simple harmonic motion is constant whereas the total energy of an oscillation, in general, needs not to be constant.
What is torsional pendulum?
a pendulum the weight of which is rotated alternately in opposite directions through a horizontal plane by the torsion of the suspending rod or spring: used for clocks intended to run a long time between windings.
What is the formula for spring stiffness?
– ‘G’ is the modulus of rigidity of the spring material. – ‘d’ is the diameter of the wire in which the helical spring is made. – ‘R’ is the radius of the helical spring. – ‘n’ is the number of turns of the total helical spring geometry.
What is the formula for spring tension?
k = Spring constant (varies depending on how “tight” the spring is. (N/m) x = change in spring length from starting position (Ex. the spring may be 2 meters long, but when you hang a 20g weight from it, it may elongate to be 3 meters long. So, in this case, x would equal 1 meter.) (m) PEs = Potential energy of the spring. (J)
How to measure the stiffness of a spring?
about the mathematical form of Hooke’s law
How to find the equivalent spring stiffness?
Spring constant or force constant k. In Hooke’s law,we find a constant k which is known as force constant or stiffness constant.