A wheel with a radius of 54.0 cm rolls without slipping along a horizontal floor.?

Question

At time t1, the dot P painted on the rim of the wheel is the point of contact between the wheel and the floor, as shown in th figure below. At a later time t2, the wheel has rolled through two and a half revolutions.
What is the magnitude of the displacement of P?
What is the angle (relative to the floor) of the displacement of P?

Answer 1

You can get the cirumference by multiplying the radius by 2 pi so that is 169.6 cm.

It rolled 2 1/2 revolutions (each being one circumference distance) so that is 169.6 x 2.5 = 424.1 cm displacement.

The final angle is 180 degrees (a half revolution) because the 2 revs do not count - they just put the dot back on the floor so only the final half rev is a visible angular change in dot position.

Answer 2

the magnitude, you first need to find out how far it has rolled;
so 2.5 revolutions x 2 x pi x 54 radius
848.2300165 which makes which you can round accordingly. This gives you the distance along the floor but P is not on the floor so you ahve to use pythagorus' theorum to work out the length of the hypotenuse. This would be square root of (848.2300165 squared + 104 squared).
I make that to be 854.5818632 cm. Remember, this is the corrct distance because displacement is the direction in a straight line in a certain direction. You must go straight from where you started to where P ends.

for the angle it is from where it started because it is the angle of the displacement. For this you have to find out the angle of elevation. P, the floor and the starting point form a triangle. from P to the floor is 108 cm (2 x the radius) and the distance along the floor is 848.2300165 cm so using sohcahtoa (i assume you understand what i mean by that - its sin, cos and tan.) tan angle is 108/ 848.2300165 so the angle is tan^-1 108/848.2300165
again, using my calculator, this comes to 7.256082911 degrees which needs rounding.
I hope you understand what I'm think there, if not try reading through it and double checking where i got my figures from and you should get it sooner or later.
good luck, Louise x

Answer 3

108 cm = displacement

After two revolutions, the dot will still be at the same point, but after another half, the dot will be across the diameter of the wheel- the displacement will be how far it is from the start- or the diameter= 2r = 108 cm.

90 degrees

The angle, I suppose, is 90 degrees since the floor and the displacement vector creates a right angle.

Answer 4

a) The dot has moved π.r in the X direction ( because that is how far the wheel has moved ) and the dot has climbed up 2r ( the diameter of the wheel ) So the displacement is √ (πr)^2 + (2r)^2 = 1.12 m Angle of displacement = tan^–1 (2r / πr ) = 32.5º X displacement = 0.942 m Y dispoacement = 0.6 m

Answer 5

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Answer 6

3 degrees.