A wheel with a radius of 45.0 cm rolls without slipping along a horizontal floor (Fig. 4-21). A

A wheel with a radius of 45.0 cm rolls without slipping...

A wheel with a radius of $45.0 \mathrm{~cm}$ rolls without slipping along a horizontal floor (Fig. 4-21). At time $t_{1}$, the dot $P$ painted on the rim of the wheel is at the point of contact between the wheel and the floor. At a later time $t_{2}$, the wheel has rolled through one-half of a revolution. What are (a) the magnitude and (b) the angle (relative to the floor) of the displacement of $P$ during this interval?

Key Concepts

Rolling Without Slipping

This concept refers to the motion where an object rolls on a surface in such a way that its point of contact with the surface is momentarily at rest. It implies a fixed relationship between the translational motion of the object’s center and its rotational motion, ensuring that the distance traveled equals the arc length described by the rotation.

Angular Displacement

Angular displacement is the measure of the angle through which an object rotates about a fixed axis. In problems involving rolling motion, the angular displacement is directly related to the arc length traversed by points on the object's rim, connecting rotation to linear movement.

Displacement of a Point on a Rotating Body

This concept deals with calculating the overall change in position of a point on a rolling object. It combines the translational displacement of the object’s center with the additional motion imparted by the rotation, often resulting in a cycloidal path that is distinct from the straight-line motion of the center.

Transcript

00:01 So this question is asking us to make our own displacement vector given a specific example of a rolling sphere.

00:07 So this is going to use all the knowledge of vectors and trigonometry so far.

00:13 So we're given that the sphere, so the point p on the sphere moves from being at the bottom of the sphere to the top of the sphere from when it rolls from a time t1 to t2.

00:25 So it undergoes a half revolution.

00:30 So we have our displacement.

00:32 Vector drawn in here is in yellow.

00:35 So then we're going to call this vector or.

00:43 So our vector or is a displacement vector.

00:46 And now we can make a right angle triangle out using this vector and the ground.

00:51 So we're going to call this side of the triangle y and this side of the triangle x.

00:57 So this is the right angle triangle.

00:59 And we're going to call this angle theta, which is the angle the vector makes with the horizontal.

01:06 So the magnitude of this vector or we're going to denote as so the magnitude of the vector or is equal to just a lowercase or and now we can tackle the question since we have all the notation out the way so we know so using this round angle triangle we can use pythagoras theorem and the trigonometric ratios so what size of this triangle do we know so we know the magnitude of the vector or so we know the length of the hypotenuse and we also know this side since we're given that the length, that the length of, or that the radius of the sphere is 0 .45 meters, which is given as 45 centimeters.

02:03 So we have the radius, so we have the diameter of this sphere and thus we have y.

02:08 So why is equal to 0 .9 meters.

02:18 So then you, so we have y now.

02:21 And we're also told that the sphere undergoes a half revolution.

02:24 So that means it rolls half the, half the length of the, its circumference...

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