Luke is trying to catch a pesky animal that keeps eating vegetables from his garden. He is building a trap and needs to use a spring to close the door to his trap. He has a spring in his garage and he wants to determine the spring constant of the spring. To do this, he hangs the spring from the ceiling and measures that it is 19.1 cm long. Then he hangs a 1.10-kg brick on the end of the spring and it stretches to 40.1 cm.
This question exist of 4 parts I finished a and b, but still need c and d! Anyhow I am not getting the right answer someone please help.
(c) When the displacement is 2.55 cm from the equilibrium position, what is the speed of the brick?
(d) How long will it take for the brick to oscillate five times?
Thanks!
The spring constant is
k = M*g/0.21 m = 51.3 N/m
(c) The sum of kinetic and (spring) potential energy is constant. The answer depends upon how much the spring was stretched before it was released and set in motion. You did not provide that information.
The oscillation period is
P = 2*pi*sqrt(M/k) = 0.92 seconds
It will take five tinmes that to osciallate five times, no matter how much it was initially stretched.
a)(a) What is the spring constant of the spring?
51.3 N/m
(b) Luke now pulls the brick 5.1 cm from the equilibrium position to watch it oscillate. What is the maximum speed of the brick?
.35 m/s
So would you subtract 2.55 cm from 5.1 cm? To get how much was stretched?
To find the solution to these questions, we need to use the principles of simple harmonic motion and the equation for the period of oscillation. Let's go step by step:
(c) To determine the speed of the brick when it is 2.55 cm from the equilibrium position, we need to find the maximum potential energy (U) of the system at that position. The maximum potential energy is given by:
U = (1/2)kx²
where k represents the spring constant and x is the displacement from the equilibrium position.
Now, to find the speed of the brick at that position, we can use the conservation of mechanical energy. At the equilibrium position, the total mechanical energy is the sum of potential energy (U) and kinetic energy (K), given by:
E = U + K
At the maximum displacement, all the potential energy is converted into kinetic energy, so the total mechanical energy (E) is equal to the kinetic energy (K):
E = K
Since kinetic energy (K) is given by:
K = (1/2)mv²
where m represents the mass of the brick and v is the speed, we can solve for v:
v = sqrt(2E/m)
Now, substituting U = (1/2)kx² into the equation for E, we have:
E = (1/2)kx²
Therefore, the speed of the brick at a displacement of 2.55 cm is:
v = sqrt(kx²/m)
Now, we can calculate it:
x = 2.55 cm = 0.0255 m
m = 1.10 kg
Substituting these values into the equation, we get:
v = sqrt(k(0.0255)²/1.10)
Now, let's move on to part (d).
(d) To calculate the time taken for the brick to oscillate five times, we need to determine the period of oscillation (T). The period is given by:
T = 2π(sqrt(m/k))
We already know the values of m and k:
m = 1.10 kg
k = spring constant (to be determined)
Now, we need to solve for the spring constant (k) using the data provided:
Using Hooke's Law:
F = kx
where F is the force exerted by the spring and x is the displacement.
At maximum displacement, the force exerted by the spring is equal to the weight of the brick, given by:
F = mg
Since we know the displacement (x) at maximum stretching (40.1 cm = 0.401 m) and the weight of the brick (mg = 1.10 kg * 9.8 m/s²), we can solve for k:
k = F/x = mg/x
Substituting the given values, we find:
k = (1.10 kg * 9.8 m/s²) / 0.401 m
Now, we have the spring constant (k), and we can substitute it into the equation for the period of oscillation (T) to find the time taken for five oscillations.
I hope this helps! Let me know if you have any further questions.