The spring shown in the image below has an unstretched length of 3.00m, and a spring constant of 75.0N/m. The mass of m=7.50kg is attached to the end of the spring, and the mass is pulled down to stretch the spring to 4.50m in length. This began the mass moving up and down with simple harmonic motion (SHM). In this simple harmonic motion;
a) What is the spring length;
i) At the locations where the velocity of the mass is zero (2)
ii) At the location where its speed is maximized (1)
iii) when its elastic energy is “0J” (1)
b) Determine the acceleration of the mass when the spring is 4.20m long.
c) Determine the speed of the mass when the spring length is 4.30m long.
d) Determine the time it will take the mass to bounce up to the maximum
height and back to its lowest height eight times.
To answer these questions, we will use the equations of motion for simple harmonic motion (SHM) and the concepts of energy conservation.
a) To find the spring length at specific points during SHM, we need to understand that the total mechanical energy (the sum of the kinetic and potential energy) remains constant throughout the motion.
i) At the locations where the velocity of the mass is zero:
When the mass is at the extreme points of its motion, its velocity is zero. At these points, the potential energy is maximized, and the spring is fully stretched or compressed. In this case, the spring length would be equal to the sum of the unstretched length and the amplitude of the motion. The amplitude is the maximum displacement from the equilibrium position.
Therefore, the spring length at these locations would be:
Spring length = Unstretched length + Amplitude
Spring length = 3.00 m + (4.50 m - 3.00 m) [Given that the spring is stretched to 4.50 m]
Spring length = 3.00 m + 1.50 m
Spring length = 4.50 m
ii) At the location where its speed is maximized:
The speed of the mass is maximized when it passes through the equilibrium position. At this point, the potential energy is zero, and all the mechanical energy is converted to kinetic energy.
Since the potential energy is zero, the spring is at its unstretched length when the speed is maximized. Therefore, the spring length would be equal to the unstretched length.
Spring length = 3.00 m
iii) When its elastic energy is “0J”:
Elastic energy is the potential energy stored in a spring due to its deformation. When the elastic energy is zero, the spring is neither compressed nor stretched, meaning it is at its unstretched length.
Spring length = 3.00 m
b) To determine the acceleration of the mass when the spring is 4.20 m long:
The acceleration can be calculated using the equation for the restoring force of a spring:
F = -kx
Where:
F is the force exerted by the spring,
k is the spring constant,
x is the displacement from the equilibrium position.
To find the acceleration, we need to find the force exerted by the spring at a displacement of 4.20 m from the equilibrium position.
Force exerted by the spring:
F = -kx
F = -(75.0 N/m)(4.20 m)
The negative sign indicates that the force is directed opposite to the displacement from the equilibrium position.
Acceleration:
a = F/m
a = [-(75.0 N/m)(4.20 m)] / 7.50 kg
Calculate the numerical value to find the acceleration.
c) To determine the speed of the mass when the spring length is 4.30 m long:
The speed of the mass can be calculated using the equation for simple harmonic motion:
v = ω√(A^2 - x^2)
Where:
v is the velocity of the mass,
ω (omega) is the angular velocity (2πf, where f is the frequency),
A is the amplitude of the motion,
x is the displacement from the equilibrium position.
To find the velocity, we need to know the values of A (amplitude) and x (displacement).
Given that the spring is stretched to 4.50 m and the length is now 4.30 m, we can find the displacement from the equilibrium position:
Displacement from equilibrium = |Length - Unstretched length|
Displacement from equilibrium = |4.30 m - 3.00 m|
Plug in the values of A and x into the velocity equation to calculate the speed.
d) To determine the time it will take the mass to bounce up to the maximum height and back to its lowest height eight times:
The time period for one complete cycle of SHM can be calculated using the formula:
T = 2π√(m/k)
Where:
T is the time period,
m is the mass attached to the spring,
k is the spring constant.
Once we know the time period, we can multiply it by the number of cycles (eight in this case) to find the total time.
Calculate the numerical value to find the time.