Student Neighbour Needs Help In Arriving At The Answer

[kuiperbelt]

Please see link below:

https://ibb.co/8bWtrM5

[bednobs]

i dont get why your student neighbour doesn't ask his teachers how to do this stuff. Surely it's better to understand the method, and that is what he(if it's HE)/we(if it's FE) are paying for?

[bednobs]

not that im miserable about it, i like looking at the q's. It just feels like this hel is already being paid for by someone but not being used

[smurfchops]

Getting tired of this person’s questions now.

[Etch]

Consider triangle OAB and
let length of OA = a
let angle ABO = A
let length of AB = b
and let angle AOB = B

For this triangle the Sine Rule says that
sin(A)/a = sin(B)/b ...... (1)

Now, a = 0.5 m, A = φ, b = 1.5 m, B = 180° - θ
and since sin(180° - θ) = sin(θ) equation (1) gives

sin(φ)/0.5 = sin(θ)/1.5

Multiply both sides by 0.5:

sin(φ) = 0.5 × sin(θ)/1.5 = sin(θ)/3 ...... (2)

When θ = 60°, sin(θ) = √3/2
therefore sin(φ) = √3/6
and φ = sin⁻¹(√3/6) ≈ 16.78°

It is a well-know property of functions of one independent variable that if f is a function of a variable u, and u is a function of a variable x, then
df/dx = df/du . du/dx
Differentiating both sides of (2) with respect to time, t, using this property gives

d(sin(φ))/dφ . dφ/dt = d(sin(θ)/3)/dθ . dθ/dt

For any x, d(sin(x))/dx = cos(x) so

cos(φ) . dφ/dt = (cos(θ)/3) . dθ/dt

Dividing both sides by cos(φ) gives

dφ/dt = (cos(θ)/3cos(φ)) . dθ/dt

When θ = 60°, cos(θ) = 0.5 and, as shown above, cos(φ) ≈ cos(16.78°) ≈ 0.9574
If dθ/dt = 10 rev/sec, multiply by 2π to convert to rad/sec giving
dθ/dt = 20π rad/sec and so

dφ/dt ≈ (0.5/(3 × 0.9574)) . 20π ≈ 10.94 rad/sec

QED

[THECORBYLOON]

SMURF, if it's getting too much for you, don't look at any future questions.

[Etch]

PS - In my answer above I neglected to explain that the "rate at which the connecting rod is turning" is given by the rate of change of angle φ, expressed mathematically as dφ/dt, and "the crank makes 10 rev/sec" means that the rate of change of angle θ, expressed mathematically as dθ/dt, is 10 times 360° per second.

[jourdain2]

Don't they ever do any arts or humanities? Equally hard questions (if not harder) abound there, but I suppose students tend not to recognise that they are hard. :(

[ZebuSanctuary]

// I neglected to explain... dθ/dt, is 10 times 360° per second //

But you didn't, look...

dθ/dt = 10 rev/sec, multiply by 2π to convert to rad/sec giving

dθ/dt = 20π rad/sec

// "rate at which the connecting rod is turning" //

It's only right, the statement above is highlighted in inverted commas to dispel any ambiguity.

Since it is quite apparent that the connecting rod isn't turning (as in completing a full revolution), but is actually describing an arc.

One wonders if this might have been a misconception held by the student such as to cause his/her confusion?

If you ask me, BA is rightfully yours :-)

[ABSpareEditor]

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[kuiperbelt]

Zebu>08.13 Fair Points!!