A mass (0.5 kg) and spring (k = 250 N/m) system oscillates with an amplitude of 3.5 cm.
a) What is the total mechanical energy of the system?
b) What is the maximum speed of the mass?
c) What is the maximum acceleration of the mass?
I need help please! Do it step by step and then substitute the numbers into that equation
Sure! Let's go step by step to find the answers to your questions.
a) To find the total mechanical energy of the system, we can use the formula:
Total mechanical energy = (1/2) * m * v^2 + (1/2) * k * x^2
where m is the mass of the system, v is the velocity, k is the spring constant, and x is the amplitude.
In this case, the given values are:
m = 0.5 kg (mass)
k = 250 N/m (spring constant)
x = 3.5 cm (amplitude)
First, let's convert the amplitude to meters:
3.5 cm = 0.035 m
Now, we need to find the maximum velocity of the mass. Since the system oscillates, we know that the energy is constantly changing between kinetic energy and potential energy. At the maximum displacement (amplitude), the energy is entirely potential energy. As the mass crosses equilibrium, it converts all potential energy into kinetic energy. Therefore, the maximum velocity occurs when the mass passes through equilibrium.
At equilibrium, the spring force is zero and only the kinetic energy exists. So, we can equate the potential energy to the kinetic energy:
(1/2) * k * x^2 = (1/2) * m * v^2
Simplifying the equation, we get:
v^2 = (k/m) * x^2
Now we can substitute the given values:
v^2 = (250 N/m) / (0.5 kg) * (0.035 m)^2
Next, we solve for v:
v = sqrt((250 N/m) / (0.5 kg) * (0.035 m)^2)
Now that we have the value of v, we can substitute it into the formula for total mechanical energy.
Total mechanical energy = (1/2) * m * v^2 + (1/2) * k * x^2
Substituting the values:
Total mechanical energy = (1/2) * (0.5 kg) * (sqrt((250 N/m) / (0.5 kg) * (0.035 m)^2))^2 + (1/2) * (250 N/m) * (0.035 m)^2
Simplifying the equation will give us the answer to part a).
b) To find the maximum speed of the mass, we have already found it in the previous step as v.
c) The maximum acceleration of the mass occurs at the extreme points (when it is at the maximum displacement). At these points, acceleration is given by the formula:
a = (k/m) * x
where a is the acceleration, k is the spring constant, m is the mass, and x is the amplitude.
Now we can substitute the given values:
a = (250 N/m) / (0.5 kg) * 0.035 m
Simplifying the equation will give us the answer to part c).
I hope this helps! Let me know if you have any further questions.
"Do it step by step and then substitute the numbers into that equation "
You are kidding.