So I'm doing a lab about simple harmonic information and I have to answer questions at the end of a lab were I'm provided this question to answer
here's a step that's not the actual question but the question makes reference to it
step 3
Using the pointer and the scale, note the equilibrium position of your mass on the spring, pull the mass down a given distance, record it, and release the mass. Record the position that the mass rises to. Do this for each mass.
Here's a question that asked me about step 3 which is not the actual question I have but my answer helps answer my question
So here was question 1
How did the distance above and below the equilibrium positions compare for each of the masses in step 3?
here's my answer
1. When we performed step three in the procedure that when we pulled the mass spring system down 4.0 cm from the equilibrium position with mass that it went past this equilibrium position 4.0 cm. For every single different mass that was used to conduct this experiment the same thing also occurred.
HERE'S MY QUESTION...
Using your experimentally determined spring constant, your largest mass, and your amplitude for that mass from step 3, calculate
a. The total eneergy of your spring-mass system;
b. the velocity of the mass when it passes through its equilibrium position (E total = E kinetic)
C. The potential energy and kinetic energy of your Simple Harmonic Oscillator when its displacement is half the amplitude. (x=A/2 E potential = 1/2 kx^2 & E total = E potential + E kinetic)
d. The velocity of the mass when it displacemnt is half the amplitude.
ok and heres my data
largest mass = .117 kg
Amplitude = 4.0 cm
100 cm = m
(4.0 cm) / (100 cm) = .04 m
here's all of my data on for this mass that I don't may help or may not help asnwer this question
Force of gravity = -.49 N
x (this distance is the x I pulled down the spring to find the observed period that I found by finding the time for 10 oscillations) = -.01 m
spring constant k = 49 N/m
T observed = .33 s
T calculated = .20s
Ok I don't know how to answer these questions... I don't know what formula to use for E total when it's not direcly given in the question like in point a
If you could please show me how to do this and all of the formulas used and proper sig figs....
Thank you for the help!
To answer the questions, we will use the formulas for calculating the total energy of a spring-mass system, the velocity of the mass at equilibrium, and the potential and kinetic energy at half the amplitude.
a. Total energy of the spring-mass system:
The total energy of a spring-mass system is the sum of the potential energy (E_potential) and the kinetic energy (E_kinetic).
E_total = E_potential + E_kinetic
The potential energy of a spring-mass system can be calculated using the formula:
E_potential = 1/2 * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
The kinetic energy of a mass can be calculated using the formula:
E_kinetic = 1/2 * m * v^2
where m is the mass and v is the velocity.
b. Velocity of the mass at equilibrium:
To find the velocity of the mass when it passes through its equilibrium position, we can use the fact that the total energy (E_total) is equal to the kinetic energy (E_kinetic):
E_total = E_kinetic
1/2 * m * v_eq^2 = E_total
1/2 * m * v_eq^2 = 1/2 * k * A^2
where v_eq is the velocity at equilibrium and A is the amplitude of the motion.
c. Potential energy and kinetic energy when displacement is half the amplitude:
To find the potential energy and kinetic energy when the displacement is half the amplitude, we can use the given equation:
E_potential = 1/2 * k * x^2
E_kinetic = 1/2 * m * v^2
where x is half the amplitude (A/2).
d. Velocity of the mass when displacement is half the amplitude:
To find the velocity of the mass when the displacement is half the amplitude, we can use the equation:
E_total = E_kinetic
1/2 * m * v_half^2 = E_total
1/2 * m * v_half^2 = 1/2 * k * (A/2)^2
where v_half is the velocity when the displacement is half the amplitude.
Using your provided data, let's calculate the answers:
a. To find the total energy (E_total), we need to find the potential energy and kinetic energy separately. Given that the amplitude is 4.0 cm (0.04 m) and the spring constant is 49 N/m, we can calculate the potential energy using:
E_potential = 1/2 * k * x^2
E_potential = 1/2 * 49 N/m * (0.04 m)^2
E_potential = 0.04 J
b. To find the velocity (v_eq) at equilibrium, we can use the equation:
1/2 * m * v_eq^2 = E_total
1/2 * (0.117 kg) * v_eq^2 = 0.04 J (as calculated in part a)
v_eq^2 = (2 * 0.04 J) / 0.117 kg
v_eq = √(2 * 0.04 J / 0.117 kg)
Calculate the value using a calculator.
c. To find the potential energy (E_potential) and kinetic energy (E_kinetic) when the displacement is half the amplitude, which is x = A/2 = 0.02 m, we use the given formulas:
E_potential = 1/2 * k * x^2
E_potential = 1/2 * 49 N/m * (0.02 m)^2
E_potential = 0.02 J
E_kinetic = 1/2 * m * v^2
E_kinetic = 1/2 * (0.117 kg) * v^2
Now we need to determine the value of v when the displacement is half the amplitude (x = 0.02 m). For this, we can use the conservation of energy:
E_total = E_kinetic + E_potential
E_total = 0.02 J (as calculated in part c)
0.02 J = 1/2 * (0.117 kg) * v^2
Solve the equation to find the value of v.
d. The velocity (v_half) of the mass when the displacement is half the amplitude (x = A/2 = 0.02 m) can be found using the equation:
1/2 * m * v_half^2 = E_total
1/2 * (0.117 kg) * v_half^2 = 0.02 J (as calculated in part c)
v_half^2 = (2 * 0.02 J) / 0.117 kg
v_half = √(2 * 0.02 J / 0.117 kg)
Calculate the value using a calculator.
Remember to include appropriate significant figures in your final answers based on the given data and the accuracy of the measurements.