A wheel with a radius of 45.0 cm rolls without slipping along a horizontal floor (Fig. 3-37 ) .

step 1 So the point P is displaced vertically by two times the radius R, which is 0 .450 meters, which

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. $3-37 ) .$ 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 $P ?$

A wheel with a radius of 45.0 $\mathrm{cm}$ rolls without slipping along a horizontal floor (Fig. $3-37 ) .$ 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 $P ?$

Key Concepts

Rolling Without Slipping

This concept ensures that the point of contact between a rolling object and the surface does not slide, meaning its instantaneous velocity is zero. This condition ties the rotational motion of the wheel directly to its translational motion, as the distance traveled by the wheel is equal to the arc length described by its rotation.

Translation and Rotation Decomposition

In the analysis of rolling motion, the movement of any point on the rolling object is viewed as the sum of the translational motion of the object’s center of mass and the rotational motion of the point around that center. This decomposition allows for separate analysis of each motion component and then their vector recombination.

Vector Displacement

Vector displacement is the overall change in position of a point, which is determined by summing its individual displacement components. It has both magnitude and direction, meaning that calculating the resultant displacement requires careful consideration of the contributions from both the horizontal and vertical paths.

Trigonometry in Displacement

Trigonometric methods are used to resolve the displacement vector into horizontal and vertical components and to determine its magnitude and direction. By applying sine, cosine, and inverse trigonometric functions, one can find the angle relative to a reference direction (such as the floor) and the corresponding length of the displacement vector.

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