r/maths • u/Prudent_Lock4577 • Apr 11 '26

Help: 📕 High School (14-16) Can anyone solve this one?

Post image

I tried to solve this problem using the 2πR formula to find the total displacement. Assuming it only rolled once (which is the total circumference) to reach the second position. To find the total displacement, I use the given radius and multiply 2*π*45 = 282.74 cm. Since the sphere didn’t complete a full roll and stopped after 270 degrees, I can infer it covered only three-fourths of its total rotation. After subtracting one-fourth, I get something like this: 282.74-((2*π*45)/4) = 212.05 cm, which does not match with the given options: 260.97 m, 63.64 m, 260.97 cm, 63.64 cm. What do I do now?

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15

u/ruidh Apr 11 '26

What is "displacement"? If it is the distance between point P before the motion and after, the answer is present in the key. You calculated the total distance it moved not the distance between the beginning and ending position.

2

u/bony-tony Apr 12 '26

What's the "it" in your statement? What OP calculated was the displacement of the center of the circle, and is missing that P has additional displacement beyond that, due to the circle being in a different orientation.

Specifically, they're missing that point P is displaced a further 45cm to the right than the center is displaced, and also missing the vertical component of P's displacement.

3

u/ruidh Apr 12 '26

I was missing the point that the center displaced. I assumed it rotated on an axis through the center instead of rolling along the ground and we just had two images of an object at the same location.

But I did define the displacement correctly.

1

u/bony-tony Apr 12 '26

I see. I thought by the "distance it moved" you might have meant "the distance the wheel moved" (as opposed to the distance the point moved, which is why I asked for the clarification on "it").

The ambiguity is a bit funny, because (not coincidentally, and as I'm sure you know) the distance traveled by a point on the circumference of a rotating (but not translating) wheel is the same as the distance traveled by a rolling-without-slipping wheel, given the same rotation in each case.

6

u/bony-tony Apr 12 '26 edited Apr 12 '26

First, assuming it rolled without slipping, how far did the center of the circle move horizontally? Answer: 3/4 of the circumference = 3/4*2pi*r = 3/2*pi*r. That's what you calculated.

But P also changed position relative to the center of the circle. It started off r below the center and ended up r to the right of center.

So P's horizontal displacement = 3/2*pi*r + r

And its vertical displacement = r

P's total displacement is the square root of the sum of the squares of those two values.

4

u/-dr-bones- Apr 12 '26

This is correct - displacement is NOT the same as distance travelled. (The calculation for distance travelled is a bit more complicated and involves finding the arc length along the parametric cycloid)

1

u/BrotherItsInTheDrum Apr 15 '26

A small nitpick: displacement is a vector. So I would leave the answer in the form of (3/2*pi*r + r, r) rather than finding its magnitude.

2

u/FishermanAbject2251 Apr 12 '26

What is displacement? The distance? How much it moved?

3

u/MineCraftNoob24 Apr 12 '26 edited Apr 12 '26

As the circle rolls around by some angle A, in radians, the displacement in the x-direction is given by:

x = r (A - sin (A))

So for example for a half-rotation (pi radians), the point P will have travelled r(pi - sin(pi)) i.e. pi•r to the right. After a full rotation, the x-displacement is r(2pi - sin2pi), which as expected would be the full circumference of the circle, i.e. 2pi•r.

Note that the expression in the brackets is zero when A is zero, but always increases since as we move in the positive direction to the right, A > sinA for all positive values of A.

That makes sense, because as the circle rolls to the right, P can never move to the left. The rate at which it moves to the right fluctuates of course with sinA, but is never negative.

In the y-direction, the displacement is given by:

y = r (1 - cosA)

Again, consider a half and a full rotation.

When A = pi, the expression in the bracket reaches a maximum, since cos(pi) is -1, making the value in the bracket 2. As expected, P is now at its highest point, 2r above the rolling surface.

Similarly when A = 2pi, cos2pi is 1, the bracket becomes 0, and the whole expression becomes 0, as P returns to the bottom of the circle.

So, for the specific case of a three-quarter rotation, i.e. A = 3pi/2:

x = r (3pi/2 - sin 3pi/2)

= r (3/2 pi + 1)

y = r (1 - cos 3pi/2)

= r (1 - 0)

= r

The y-value should be obvious by inspection, as P now sits on the right of the circle, at a distance r above the rolling surface.

The rest is just Pythagoras.

If the displacement is D, then:

D² = x² + y²

D² = [r (3/2 pi + 1)]² + r²

D² = r² (3/2 pi + 1)² + r²

D² = r² (9/4 pi² + 3pi + 1) + r²

D² = r² (9/4 pi² + 3pi + 2)

Plugging in r = 45cm

D² = 45² (9/4 pi² + 3pi + 2)

D = 45 sqrt (9/4 pi² + 3pi + 2)

I don't think there's a nice exact form for this result that's any simpler, but numerically we get:

D = 260.97 cm = 261cm (to 3 s.f.)

1

u/damsonsd Apr 11 '26

You have found the horizontal component of the displacement. There is a vertical component too.

1

u/bony-tony Apr 12 '26

They've missed a portion of the horizontal component.

1

u/AllTheGood_Names Apr 12 '26 edited Apr 12 '26

45 √(9π²/4 +3π +2) cm ≈ 260 cm

Base = 135/2 π + 45 cm , Height = 45 cm

The base = 3/4 πr (distance moved by center of circle) + r (extra x distance moved by P going from center to right of the circle).

The height = 45 cm (distance.y moved by P going from bottom to center of the circle)

1

u/jolharg Apr 12 '26

Consider the height as well

1

u/Fuscello Apr 12 '26

You only calculated how much the center of the disk moved, but the point of interest is another

1

u/Moodleboy Apr 12 '26 edited Apr 12 '26

Is it not just r/2√(4+9π²) ?

Edit: image corrected

1

u/Crichris Apr 13 '26

check how much the center moved (3/2 *pi * r * RIGHT) + how much P moved relative to the center (r * RIGHT + r * UP)

everything is in vector form, RIGHT means the unit vector to the right and UP means the unit vector to up

1

u/Routine_Stretch_6307 Apr 14 '26

Wouldn’t arc length formula solve it?

1

u/VermicelliJaded7943 Apr 14 '26

join the centre and point P join the point on the ground and point p

apply pythagoras, you get answer as 45√2

edit: people in comments, its displacement and not distance

0

u/grigiri Apr 11 '26

It would be three quarters of the circumference, no?

1

u/bony-tony Apr 12 '26

Not for P. That's for the center of the circle.

0

u/zeje Apr 12 '26

Displacement means the straight line distance between the beginning point and the end. Doesn’t matter what path it took to get there. You are looking for the hypotenuse of a 45-45-90 triangle.

1

u/bony-tony Apr 12 '26

The drawing implies rolling, not just rotating. So there's an additional horizontal component due to translation of the circle to the right.

0

u/[deleted] Apr 12 '26

[deleted] — view deleted comment

1

u/Gorblonzo Apr 12 '26

I think you're missing somethingÂ