A 492g oscillator has a speed of 99.5 cm/s when its displacement is 3.60 cm and 72.5 cm/s when its displacement is 5.10 cm.
What is the oscillator's maximum speed?
I'm suck with this problem, can someone please go through it with me step by step, please and thanks
Total energy (kinetic plus potential energy) at any point in the oscillator is a constant.
KE = (1/2)m v^2
PE = (1/2)kx^2
TE = KE + PE
KE1 + PE1 = KE2 + PE2
0.2435J + PE(3.60cm) = 0.1293J + PE(5.10cm)
0.1142J = PE(5.10cm) - PE(3.60cm)
0.1142J = (1/2) k(5.10cm)^2 - (1/2) k (3.60cm)^2
0.1142J = (1/2) k ((5.10cm)^2 - (3.60cm)^2))
0.2284J = k (13.05cm^2)
k = 175.0 kg/s^2
PE(3.60cm) = (1/2) (175.0kg/s^2) (3.60cm)^2
PE(3.60cm) = 0.1134J
TE = KE(3.60cm) + PE(3.60cm)
TE = 0.2435J + 0.1134J
TE = 0.3569J
At maximum speed, all the energy is kinetic.
KE = TE
(1/2) m v^2 = TE
v^2 = 2 TE / m
v = sqrt(2 TE / m)
v = sqrt( 2 0.3569J / 492g)
v = sqrt(1.451 m^2/s^2)
v = 120cm/s
Well, well, well, looks like we have a little physics puzzle on our hands! Don't worry, I'm here to clown around and help you out step by step! Let's get started.
We're given the mass of the oscillator, which is 492 grams, but we'll need that in kilograms for our calculations. So, let's convert that mass to kilograms. One kilogram is equal to 1000 grams, so we divide 492 by 1000:
492 grams / 1000 = 0.492 kilograms
Now, let's focus on the speed and displacement values. We have two different values for speed and displacement:
When the displacement is 3.60 cm, the speed is 99.5 cm/s.
When the displacement is 5.10 cm, the speed is 72.5 cm/s.
To find the maximum speed, we need to figure out the point where the speed is the highest. In other words, we need to find the point where the displacement is the greatest.
So, we know that the displacement is 3.60 cm when the speed is 99.5 cm/s, and the displacement is 5.10 cm when the speed is 72.5 cm/s.
Now, let's compare these two displacements. 3.60 cm is smaller than 5.10 cm, so we can conclude that the highest speed must occur at the displacement of 5.10 cm. This will be our maximum speed.
Therefore, the oscillator's maximum speed is 72.5 cm/s.
And there you have it! Step by step, we've solved the problem and found the maximum speed. Always remember, physics can be a circus, but with a little clowning around, we can make it fun!
Sure! Let's solve this problem step by step.
Step 1: Write down the given information.
We are given the mass of the oscillator (492g) and its respective speeds at two different displacements.
Mass (m) = 492g
Displacement 1 (x1) = 3.60 cm
Speed at displacement 1 (v1) = 99.5 cm/s
Displacement 2 (x2) = 5.10 cm
Speed at displacement 2 (v2) = 72.5 cm/s
Step 2: Convert mass to kilograms.
To work with SI units, we need to convert the mass from grams to kilograms.
Mass (m) = 492g = 0.492kg
Step 3: Write down the formula for velocity.
The velocity of an oscillator can be determined using the formula:
v = ω * A
Where:
v is the velocity
ω is the angular frequency (in radians per second)
A is the amplitude or maximum displacement
Step 4: Find the angular frequency for the given displacements.
For each displacement, we need to calculate the angular frequency (ω) using the formula:
ω = 2π * f
Where:
ω is the angular frequency
π is a mathematical constant approximately equal to 3.14
f is the frequency
Step 5: Find the frequency for the given displacements.
To calculate the frequency, we can use the formula:
f = v / λ
Where:
f is the frequency
v is the velocity
λ is the wavelength
Step 6: Calculate the wavelength for each displacement.
The wavelength (λ) can be determined using the formula:
λ = 2 * A
Where:
λ is the wavelength
A is the amplitude or maximum displacement
Step 7: Calculate the angular frequency for each displacement.
Using the values determined in steps 5 and 6, we can calculate the angular frequency (ω) for each displacement.
ω1 = 2π * f1
ω2 = 2π * f2
Step 8: Find the amplitude using the given displacements.
Using the formula from step 6, we can find the amplitude (A) for the given displacements.
A1 = x1 / 2
A2 = x2 / 2
Step 9: Calculate the velocity at maximum displacement.
To find the maximum velocity (vmax), we can use the equation:
vmax = √(ω^2 * A^2 + (δω/δt)^2)
Where:
vmax is the maximum velocity
ω is the angular frequency
A is the amplitude or maximum displacement
δω/δt is the rate of change of angular frequency
However, since we only have two data points, we cannot determine the rate of change of angular frequency. So, we will use an approximation method.
Step 10: Calculate the maximum velocity using the approximation method.
Using the velocities at each displacement (v1 and v2), we can calculate the maximum velocity (vmax) using an approximation:
vmax = (v1 + v2) / 2
Step 11: Convert the maximum velocity back to cm/s (if needed).
If necessary, convert the maximum velocity to the desired unit. In this case, we already have the velocities in cm/s.
So, the maximum velocity of the oscillator is the value obtained in Step 10. Plug in the numbers and calculate!
Sure! Let's break down the problem step by step.
Step 1: Understand the problem
We are given information about an oscillator's displacement at two different speeds. We need to find the oscillator's maximum speed.
Step 2: Analyze the given information
We are given two sets of data:
- When the displacement is 3.60 cm, the speed is 99.5 cm/s.
- When the displacement is 5.10 cm, the speed is 72.5 cm/s.
Step 3: Use the formula for velocity
The formula for velocity can be written as:
velocity = displacement / time
However, in this case, we are given the displacement and speed, not the time. To get around this, we can assume that the time it takes to travel a certain displacement is the same for both cases. Therefore, we can set up the following equation:
velocity1 = displacement1 / time
velocity2 = displacement2 / time
where velocity1 is the speed when the displacement is 3.60 cm, displacement1 is 3.60 cm, velocity2 is the speed when the displacement is 5.10 cm, and displacement2 is 5.10 cm.
Step 4: Solve the equation
To solve the equation, we can rearrange it to solve for time:
time = displacement1 / velocity1 = displacement2 / velocity2
Plugging in the values from our problem:
time = 3.60 cm / 99.5 cm/s = 5.10 cm / 72.5 cm/s
Calculating these values will give us the time it takes to travel the given displacements.
Step 5: Calculate the maximum speed
The maximum speed of the oscillator occurs when the displacement is at its maximum. In this case, the maximum displacement is 5.10 cm. We can use the formula velocity = displacement / time to find the maximum speed.
maximum speed = maximum displacement / time
Plugging in the values:
maximum speed = 5.10 cm / (time calculated in Step 4)
Solving this equation will give us the maximum speed of the oscillator.
I hope this step-by-step explanation helps you solve the problem! If you have any further questions, feel free to ask.