An object with mass 3.0 kg is attached to a spring with spring stiffness constant k = 300 N/m and is executing simple harmonic motion. When the object is 0.020 m from its equilibrium position, it is moving with a speed of 0.55 m/s.
(a) Calculate the amplitude of the motion.
m
(b) Calculate the maximum velocity attained by the object. [Hint: Use conservation of energy.]
m/s
Total energy (which is conserved)
= (1/2)M*V^2 + (1/2)k X^2
= 1.5*0.55^2 + 150*0.02^2 = 0.5138 J
(a) when V = 0 you get the maximum deflection Xmax. That is the amplitude.
0.5138 = (300/2)*Xmax^2
Xmax = 0.0585 m
(b) when X = 0 you have maximum speed:
0.5138 = 1.5*Vmax^2
Vmax = 0.5853 m/s
I have kids in my basement 🤪
(a) Why did the object bring a spring to the party? Was it spring break? Anyway, to calculate the amplitude of the motion, we can use the formula:
Amplitude = distance from equilibrium position = 0.020 m
So, the amplitude of the motion is 0.020 m. Just don't let the object boogie too hard!
(b) Ah, the maximum velocity. Time to unleash the energy conservation party trick! Since energy is conserved, we can equate the potential energy when the object is at its maximum displacement to the kinetic energy when it's at its maximum velocity.
At maximum displacement, the potential energy is maxed out, so it can be given by:
Potential energy = (1/2)kx^2
Where k is the spring stiffness constant and x is the maximum displacement (amplitude).
At maximum velocity, the kinetic energy is partying hard, so it can be given by:
Kinetic energy = (1/2)mv^2
Where m is the mass of the object and v is the maximum velocity.
Setting them equal to each other, we have:
(1/2)kx^2 = (1/2)mv^2
Substituting the given values:
(1/2)(300 N/m)(0.020 m)^2 = (1/2)(3.0 kg)(v^2)
Simplifying, we find:
v^2 = 0.06 m^2/s^2
Taking the square root:
v ≈ ±0.2449 m/s
Remember, the velocity can be positive or negative depending on the direction of motion. So, the maximum velocity attained by the object is approximately 0.2449 m/s. Now let's hope it doesn't lose its balance!
To solve part (a) and find the amplitude of the motion, we can use the relationship between the amplitude (A) and the maximum displacement (x_max) from the equilibrium position. The maximum displacement is given as 0.020 m.
In simple harmonic motion, the displacement (x) can be described by the equation:
x = A * sin(ωt)
Where A is the amplitude and ω is the angular frequency. The angular frequency can be calculated using:
ω = sqrt(k / m)
where k is the spring stiffness constant and m is the mass of the object.
To find the amplitude, we need to rearrange the equation as follows:
A = x_max / sin(ωt)
Now we can plug in the given values:
x_max = 0.020 m
k = 300 N/m
m = 3.0 kg
First, let's calculate the angular frequency ω:
ω = sqrt(300 N/m / 3.0 kg) = sqrt(100 rad/s^2) = 10 rad/s
Now we can calculate the amplitude:
A = 0.020 m / sin(10 rad/s * t)
Since the time (t) is not specified in the question, we cannot determine the exact value of the amplitude without knowing the specific time during the motion.
Moving on to part (b), tofind the maximum velocity attained by the object, we can use the conservation of energy. In simple harmonic motion, the total mechanical energy is conserved as the object oscillates.
The total mechanical energy is given by:
E = (1/2) * k * A^2
where E is the total mechanical energy, k is the spring stiffness constant, and A is the amplitude.
The maximum potential energy is equal to the total mechanical energy, so we can write:
E = P.E(max) = (1/2) * k * A^2
The maximum kinetic energy is reached when the displacement is zero (at equilibrium), so we can write:
K.E(max) = E - P.E(max)
Since the total mechanical energy is constant, we can set the kinetic energy equal to the potential energy at maximum displacement:
(1/2) * m * v^2 = (1/2) * k * A^2
where m is the mass of the object and v is the maximum velocity.
Now we can solve for v:
v^2 = k * A^2 / m
v = sqrt(k * A^2 / m)
Plugging in the given values:
k = 300 N/m
A (unknown)
m = 3.0 kg
Again, since the amplitude (A) is not specified in the question, we cannot determine the exact value of the maximum velocity without knowing the specific value of the amplitude.