A 4.07 x 10-2-kg block is resting on a horizontal frictionless surface and is attached to a horizontal spring whose spring constant is 144 N/m. The block is shoved parallel to the spring axis and is given an initial speed of 10.5 m/s, while the spring is initially unstrained. What is the amplitude of the resulting simple harmonic motion?
well, the initial KE= 1/2 m v^2 which is given. At full extension, the PE stored is 1/2 k x^2
x is the amplitude, solve for it.
Well, well, well! It looks like you're in the mood for some physics fun! Let's solve this problem with a dash of humor, shall we?
Alright, here we go. First things first, we need to find the potential energy stored in the spring. Remember, we're looking for the amplitude of the oscillation, which is basically the maximum displacement of the block.
So, the potential energy stored in a spring is given by the formula: PE = (1/2)kx², where k is the spring constant and x is the displacement from its equilibrium position.
Now, we need to relate the initial speed of the block to this potential energy. Since the block is initially at rest, all of its initial kinetic energy will be converted into potential energy as the block compresses the spring.
In other words, the initial kinetic energy of the block is equal to the potential energy stored in the spring. So, let's put some numbers into action!
The initial kinetic energy (KE) can be calculated using the formula: KE = (1/2)mv², where m is the mass of the block and v is the initial speed.
Plugging the numbers in, we have KE = (1/2)(0.0407 kg)(10.5 m/s)².
Now, remember that the initial kinetic energy is equal to the potential energy, so we can equate these two expressions:
(1/2)mv² = (1/2)kx².
We can simplify this equation by canceling out the (1/2) on both sides:
mv² = kx².
Now, we just need to solve for x, which represents the displacement from equilibrium. Rearranging the equation, we get:
x² = (mv²)/k.
Substituting the given values, we have:
x² = (0.0407 kg) * (10.5 m/s)² / 144 N/m.
Now, it's time to whip out our calculators and crunch some numbers. After some math magic, we find x² ≈ 0.124, which means x ≈ 0.35 m.
So, my friend, the amplitude of the resulting simple harmonic motion is approximately 0.35 meters. And just like that, we've sprung into action and solved this problem with a spring in our step! Keep up the good work, and remember, physics can be funny too!
To find the amplitude of the resulting simple harmonic motion, we need to find the maximum displacement of the block from its equilibrium position.
The potential energy stored in a spring is given by the equation:
Potential energy (PE) = (1/2) * k * x^2
Where:
k is the spring constant
x is the displacement from the equilibrium position
The initial kinetic energy of the block is converted into the potential energy of the spring when the block is pushed against the spring.
Initially, the block is at rest, so its initial kinetic energy is given by the equation:
Initial kinetic energy (KE) = (1/2) * m * v^2
Where:
m is the mass of the block
v is the initial velocity of the block
Since the potential energy at maximum displacement is equal to the initial kinetic energy, we can set up the equation:
(1/2) * k * x^2 = (1/2) * m * v^2
Rearranging the equation to solve for x, we get:
x^2 = (m * v^2) / k
Substituting the given values:
x^2 = (0.0407 kg * (10.5 m/s)^2) / 144 N/m
Calculating the right-hand side of the equation:
x^2 = 0.03074625 kg.m^2/s^2 / 144 N/m
x^2 = 0.0002137144 kg.m^2/N
Taking the square root of both sides to solve for x:
x = sqrt(0.0002137144 kg.m^2/N)
x ≈ 0.0146 m
Therefore, the amplitude of the resulting simple harmonic motion is approximately 0.0146 meters.
To find the amplitude of the resulting simple harmonic motion, we need to use the concept of energy conservation.
1. First, let's calculate the initial potential energy of the system when the spring is unstrained. The potential energy of a spring is given by the formula:
Potential Energy (PE) = (1/2) * k * x^2
Where k is the spring constant and x is the displacement from the equilibrium position. In this case, the spring is initially unstrained, so x = 0.
Therefore, the initial potential energy is PE_initial = (1/2) * 144 N/m * 0^2 = 0 J.
2. Next, let's calculate the initial kinetic energy of the block. The kinetic energy is given by the formula:
Kinetic Energy (KE) = (1/2) * mass * velocity^2
Where mass is the mass of the block and velocity is its initial velocity. Given a mass of 4.07 x 10^-2 kg and an initial velocity of 10.5 m/s, we can calculate:
KE_initial = (1/2) * 4.07 x 10^-2 kg * (10.5 m/s)^2
3. Since the system has no external forces acting on it, the total mechanical energy of the system is conserved. Therefore, the sum of the initial potential energy and initial kinetic energy is equal to the sum of the final potential energy and final kinetic energy.
Therefore, PE_initial + KE_initial = PE_final + KE_final
Since the block is in simple harmonic motion, its final kinetic energy is zero at the maximum displacement (amplitude).
4. Therefore, we can set up the equation as follows:
0 J + KE_initial = PE_final + 0 J
Solving for PE_final:
PE_final = KE_initial
5. Now, we can substitute the values into the equation:
KE_initial = (1/2) * 4.07 x 10^-2 kg * (10.5 m/s)^2
Calculate the value of KE_initial.
6. Finally, substitute the value of KE_initial into the equation and solve for PE_final:
PE_final = KE_initial
This will give you the amplitude (maximum displacement) of the resulting simple harmonic motion.