Simple Harmonic Oscillation, formula identification using UNITS.

[arc5005]

LINK TO PASSAGE: <--Click me.

At what position x will the speed of the mass be half its maximum speed, v_max?

ANSWER CHOICES <-- CLICK ME.

D) D

A fast way to solve formula-identification problems is to look at units. k has units of N/M or kg/sec2, from remembering F = kx. Looking at the choices, three have an m/k in them. The units of m/k are sec2. Since we need units of meters (units of x), this m/k must combine with the vmax so as to cancel out the sec2. This suggests that the m/k should be under a square root sign. Testing the two resulting possibilities shows that only choice D has the correct units.


Can anyone help me understand this please?

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[Lawpy]

So formula problems can be solved quickly by checking the units for each of the answer choices rather than actually deriving the formulas.

x = position, so units are meters (m)
vmax = maximum speed, so units are meters per second (m/s)
k = spring constant, so units are newtons per meter (N/m)
m = mass, so units are kilograms (kg)

Working with units is easier if we can convert everything into the simplest possible units, so only kilograms, meters and seconds. A newton is the same as kg * m/s^2 (remember Newton's Second Law: F = ma). So:

1 N = 1 kg * m/s^2 -->
1 N/m = 1 kg * m/s^2 * 1/m -->
1 N/m = 1 kg/s^2 -->
1 N/m = 1 kg * s^-2 (using negative exponents is easier to visualize: remember that a^-1 = 1/a)

Now look at each of the answer choices:

Choice A): x = vmax * sqrt(4k/m). Ignore all constants, since they don't have units (dimensionless). k/m means you are dividing the spring constant by the mass, so we get (1 N/m) / kg --> 1 kg * s^-2 * kg^-1 = 1 s^-2. Taking the square root of that gets the unit s^-1 (inverse seconds). Multiplying s^-1 with m/s results in m/s^2, which is the units for acceleration, not position. This choice is wrong.

Choice B): x = sqrt((3*m*vmax)/k). Ignore the square root for now. m*vmax is the same as kg * m/s. k has the units of 1 kg * s^-2 but because k is in the denominator, the units of 1/k is just 1 kg^-1 * s^2 (note how the exponents change sign from taking the reciprocal). Multiplying both expressions (because dividing by k is the same as multiplying 1/k), we get kg * m/s * kg^-1 * s^2 --> m/s * s^2 --> m * s. Taking the square root results in the weird units of sqrt(m * s). This choice is definitely wrong.

Choice C): x = (m/2k) * vmax. Ignore the constants. Plugging in the units, we get kg / (kg * s^-2) * m / s --> kg * kg^-1 * s^2 * m/s --> m * s. Note that this choice is similar to squaring the equation in Choice B). m * s isn't the units for position, so the choice is wrong.

Choice D): x = vmax * sqrt(3m/4k). Ignore the constants. Plugging in the units, we get m / s * sqrt(kg / (kg * s^-2)) --> m/s * sqrt(kg * kg^-1 * s^2) --> m/s * sqrt(s^2) --> m/s * s --> m, which is the unit for position, so it is the correct answer.

 

[arc5005]

So formula problems can be solved quickly by checking the units for each of the answer choices rather than actually deriving the formulas.

x = position, so units are meters (m)
vmax = maximum speed, so units are meters per second (m/s)
k = spring constant, so units are newtons per meter (N/m)
m = mass, so units are kilograms (kg)

Working with units is easier if we can convert everything into the simplest possible units, so only kilograms, meters and seconds. A newton is the same as kg * m/s^2 (remember Newton's Second Law: F = ma). So:

1 N = 1 kg * m/s^2 -->
1 N/m = 1 kg * m/s^2 * 1/m -->
1 N/m = 1 kg/s^2 -->
1 N/m = 1 kg * s^-2 (using negative exponents is easier to visualize: remember that a^-1 = 1/a)

Now look at each of the answer choices:

Choice A): x = vmax * sqrt(4k/m). Ignore all constants, since they don't have units (dimensionless). k/m means you are dividing the spring constant by the mass, so we get (1 N/m) / kg --> 1 kg * s^-2 * kg^-1 = 1 s^-2. Taking the square root of that gets the unit s^-1 (inverse seconds). Multiplying s^-1 with m/s results in m/s^2, which is the units for acceleration, not position. This choice is wrong.

Choice B): x = sqrt((3*m*vmax)/k). Ignore the square root for now. m*vmax is the same as kg * m/s. k has the units of 1 kg * s^-2 but because k is in the denominator, the units of 1/k is just 1 kg^-1 * s^2 (note how the exponents change sign from taking the reciprocal). Multiplying both expressions (because dividing by k is the same as multiplying 1/k), we get kg * m/s * kg^-1 * s^2 --> m/s * s^2 --> m * s. Taking the square root results in the weird units of sqrt(m * s). This choice is definitely wrong.

Choice C): x = (m/2k) * vmax. Ignore the constants. Plugging in the units, we get kg / (kg * s^-2) * m / s --> kg * kg^-1 * s^2 * m/s --> m * s. Note that this choice is similar to squaring the equation in Choice B). m * s isn't the units for position, so the choice is wrong.

Choice D): x = vmax * sqrt(3m/4k). Ignore the constants. Plugging in the units, we get m / s * sqrt(kg / (kg * s^-2)) --> m/s * sqrt(kg * kg^-1 * s^2) --> m/s * sqrt(s^2) --> m/s * s --> m, which is the unit for position, so it is the correct answer.

Thank you. This still seems like a ridiculous amount of work for a units type of question for a MCAT question though. I would not be able to do this quickly.

 

[Lawpy]

Thank you. This still seems like a ridiculous amount of work for a units type of question for a MCAT question though. I would not be able to do this quickly.

If this is from TBR, these calculations are way more advanced and time consuming than what you will see on practice tests and on test day. Just focus on understanding the concepts.

 

[arc5005]

If this is from TBR, these calculations are way more advanced and time consuming than what you will see on practice tests and on test day. Just focus on understanding the concepts.

Thanks. Yeah from TBR.

I will admit some of these math concepts are being "re-learned," because it has been so long since I took my last math course.

 

[Lawpy]

Thanks. Yeah from TBR.

I will admit some of these math concepts are being "re-learned," because it has been so long since I took my last math course.

Yeah and that's where practicing from TBR definitely helps. I would do them untimed. These calculations aren't easy at all and TBR deliberately picks the most complicated numbers and convoluted formulas to make sure you understand how to do calculations by hand. TBR calculations are painful, so it's good for understanding concepts thoroughly.

TBR has the most difficult passages and questions. Taking the time to learn and master the concepts deeply now will pay off huge returns on the real deal. Which is why it's important to focus on the concepts.

There are probably faster ways to do the calculations but I wanted to be thorough just so you can understand the reasoning properly. Some of the TBR solutions are vague or make too many quick and unstated assumptions that it's difficult to understand the reasoning.

 

[arc5005]

Yeah and that's where practicing from TBR definitely helps. I would do them untimed. These calculations aren't easy at all and TBR deliberately picks the most complicated numbers and convoluted formulas to make sure you understand how to do calculations by hand. TBR calculations are painful, so it's good for understanding concepts thoroughly.

TBR has the most difficult passages and questions. Taking the time to learn and master the concepts deeply now will pay off huge returns on the real deal. Which is why it's important to focus on the concepts.

There are probably faster ways to do the calculations but I wanted to be thorough just so you can understand the reasoning properly. Some of the TBR solutions are vague or make too many quick and unstated assumptions that it's difficult to understand the reasoning.

Thank you. I appreciate it!