Hi, I'm learning about simple harmonic motion of a mass-spring system and I've been working at 2 problems for the past hour. I believe I have the right formula and I know what the answers should be but Im getting a different answer. Can you explain it step to step to me please? Thanks!
When a mass 25g is attached to a certain spring, it makes 20 complete vibrations in 4s. What is the spring constant of the spring? Formula: K=4pi^2/T^2xm.
A 125N object vibrates with a period of 3.56s when hanging from a spring. What is the spring constant of the spring? Same formula mass=12.74
kg.
Answer:
For the first problem, the spring constant can be calculated using the formula K=4pi^2/T^2xm, where T is the period of the vibration (4s) and m is the mass (25g). Plugging in the values, we get K = 4*pi^2/4^2*0.025 = 39.47 N/m.
For the second problem, the spring constant can be calculated using the same formula K=4pi^2/T^2xm, where T is the period of the vibration (3.56s) and m is the mass (12.74kg). Plugging in the values, we get K = 4*pi^2/3.56^2*12.74 = 5.45 N/m.
Sure! I'll be glad to help you solve these problems step by step.
Problem 1:
Given:
Mass (m) = 25g = 0.025kg
Number of complete vibrations (n) = 20
Time (T) = 4s
We can use the formula K = (4π^2/T^2) * m to calculate the spring constant (K).
Step 1: Convert the mass to kilograms
Since the given mass is in grams, we need to convert it to kilograms by dividing it by 1000.
m = 0.025kg
Step 2: Calculate the period (T) per vibration
To find the period per vibration, divide the total time (T) by the number of vibrations (n).
T = 4s / 20 = 0.2s
Step 3: Calculate the spring constant (K)
Use the formula K = (4π^2/T^2) * m to find the spring constant.
K = (4 * π^2) / (0.2s)^2 * 0.025kg
K ≈ 98.15 N/m
Therefore, the spring constant of the spring is approximately 98.15 N/m.
Problem 2:
Given:
Mass (m) = 125N
Period (T) = 3.56s
Using the same formula K = (4π^2/T^2) * m, we can find the spring constant (K).
Step 1: Calculate the mass (m)
The given value is the force acting on the object (125N), not the mass. To calculate the mass, divide the force by the gravitational acceleration (g).
Mass (m) = 125N / 9.8m/s^2 ≈ 12.76kg
Step 2: Calculate the spring constant (K)
Using the formula K = (4π^2/T^2) * m, we can find the spring constant by substituting the values:
K = (4 * π^2) / (3.56s)^2 * 12.76kg
K ≈ 177.70 N/m
Therefore, the spring constant of the spring is approximately 177.70 N/m.
I hope this helps! Let me know if you have any further questions.
Sure! I'd be happy to help you with these problems step-by-step. Let's start with the first problem:
Problem 1:
A mass of 25g is attached to a certain spring, and it makes 20 complete vibrations in 4s. We need to find the spring constant of the spring. The formula we'll use is K = 4π^2 / (T^2 * m), where K is the spring constant, T is the period of oscillation, and m is the mass.
Step 1: Convert the mass to kilograms.
The given mass is 25g. We need to convert it to kilograms by dividing it by 1000.
m = 25g / 1000 = 0.025kg
Step 2: Calculate the period.
The problem states that the mass completes 20 vibrations in 4s. To find the period, we divide the total time by the number of vibrations.
T = Total time / Number of vibrations
T = 4s / 20 = 0.2s
Step 3: Plug the values into the formula and solve for K.
K = 4π^2 / (T^2 * m)
K = 4 * π^2 / (0.2^2 * 0.025)
K = 4 * π^2 / (0.04 * 0.025)
K = 4 * π^2 / 0.001
K ≈ 125663.706
So, the spring constant for this system is approximately 125,663.706 N/m.
Now, let's move on to the second problem:
Problem 2:
An object with a weight of 125N vibrates with a period of 3.56s when hanging from a spring. We need to find the spring constant of the spring. We can use the same formula as before, K = 4π^2 / (T^2 * m), but instead of the mass, we are given the weight.
Step 1: Convert the weight to mass.
The formula uses the mass, so we need to convert the weight to mass by dividing by the acceleration due to gravity (g ≈ 9.8m/s^2).
Weight = mass * g
125N = mass * 9.8m/s^2
mass = 125N / 9.8m/s^2 ≈ 12.76kg
Step 2: Calculate the period.
The problem states the period is 3.56s. We don't need any additional calculations for this step.
Step 3: Plug the values into the formula and solve for K.
K = 4π^2 / (T^2 * m)
K = 4 * π^2 / (3.56^2 * 12.76)
K = 4 * π^2 / (12.6736 * 12.76)
K = 4 * π^2 / 161.697536
K ≈ 0.077
Therefore, the spring constant for this system is approximately 0.077 N/m.
I hope this explanation helps! Let me know if you have any further questions.