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All ramps are not equal

I have a very tiny block that slides down a ramp in the Cartesian plane from point \(a\) to point \(b\). This ramp can be modelled by the function \(f(x)\) for \(x_a\leq x \leq x_b\). For all values of c such that \(a<c \leq b\), \(f(c)<f(a)\). If the block manages to slide from point \(a\) to \(b\). Prove that the maximum velocity of the block at point \(b\) is achieved when \(f(x)\) is a straight line from \(a\) to \(b\) if friction is present.

I will post the proof for this if no one has within 48 hours.

Note by Trevor Arashiro
7 months ago

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As the end points are fixed and are separated by a verical distance of \(h\) (say), by work-kinetic energy theorem (and a little simplification), we have, \[\frac{v_b^2}{2}=gh-\mu_kg s \] where \(\mu_k\) is the coefficient of kinetic friction and \(s\) is the length of the horizontal section of the ramp.

As the points \(a\) and \(b\) are fixed, so is \(s\). Hence, speed of the block remains constant.

PS: This was a nice fallacy which our teacher had discussed.

@Trevor Arashiro Deeparaj Bhat · 7 months ago

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@Deeparaj Bhat What if slope of the curve is less than or equal to \(\mu_{k}\)? Kushal Patankar · 6 months, 4 weeks ago

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@Kushal Patankar The question states that the body manages to slide to point \(b\). So, kinetic friction will always act.

Plus, your objection makes sense if you said that \(\mu_s \geq \tan \theta\). The inequality \(\mu_k \geq \tan \theta\) makes no difference to the motion of the block. Deeparaj Bhat · 6 months, 4 weeks ago

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@Deeparaj Bhat That's the same start to my proof, but when you say horizontal section of the ramp, are you referring only to the length of the ramp in the x-direction or the total length of the ramp? And another thing to think about: at different angles, while u is constant, the frictional force is greater for flatter sections of the ramp, so does your equation still work for curved ramps? Trevor Arashiro · 7 months ago

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@Trevor Arashiro I meant the length of the horizontal section only.

Yes, this is valid for curved ramps as well. Here's the proof:

Consider a point on the ramp. Let the tengent at this point make an angle \(\theta\) with the horizontal. Then, if \(dW\) denotes the work done by friction over a path length \(ds\), \[dW=-\mu_kmg\cos \theta ds\] But, note that \(ds\, \cos \theta=dx\). Hence, \[dW=-\mu_kmg dx\\\implies W=-\mu_kmg(x_b-x_a)\] Assuming \(x_b>x_a\)

So, all ramps are actually equal ;) Deeparaj Bhat · 7 months ago

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@Deeparaj Bhat It must be "\( \mu_k g s\)" Kushal Patankar · 7 months ago

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@Kushal Patankar Yeah. My bad. Deeparaj Bhat · 7 months ago

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Not too difficult of a proof but it does require calc (at least the way I did it... which I pray is right) Trevor Arashiro · 7 months ago

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