An object is attached to a horizontal spring. It is initially displaced by distance Δx from the equilibrium position and then released from rest. When the object passes the equilibrium position, its speed will be:
a)v = 0 m/s
b)v proportional to Δx (the speed is proportional to the initial displaceent)
c)v proportional to Δx2 (speed is proportional to the square of the initial displacement)
d)v is a non-linear function of the Δx, but cannot be written in the form "v proportional to Δx2"
My ans:
I know that a springs equilibrium position is obtained by:
Ee = 1 / 2kx^2
and elastic potential energy is related to kinetic energy via Ee=-Ek
So I solve for k in the elastic potential energy and find out how the velocity is related? I think the answer is a, but i m not sure
To determine the relationship between the speed of the object and the initial displacement, we can use the principles of simple harmonic motion. When an object is attached to a horizontal spring and released, it undergoes simple harmonic motion, oscillating back and forth around the equilibrium position.
The equation for the potential energy stored in the spring is given by:
Ee = 1/2 k Δx^2,
where Ee is the elastic potential energy, k is the spring constant, and Δx is the displacement from the equilibrium position.
As the object is released from rest, the potential energy is converted into kinetic energy when it passes through the equilibrium position. At this point, the object possesses maximum kinetic energy and no potential energy. According to the principle of conservation of energy, the mechanical energy remains constant throughout the motion.
Since the kinetic energy is proportional to the square of the velocity (Ek = 1/2 mv^2), we can deduce that the object's speed will depend on the initial displacement.
In simple harmonic motion, the maximum speed of the object is attained when it passes through the equilibrium position. Therefore, when the object passes the equilibrium position, its speed will be directly proportional to the initial displacement, Δx.
In this case, the correct answer is (b) v proportional to Δx, which means that the speed is directly proportional to the initial displacement.