Problem:
A wheel of radius r rolls without slipping on a horizontal surface shown below. If the velocity of point P is 10 m/s in the horizontal direction, the velocity of point Q (in m/s) is _______.a)0b)10c)20d)5

Solution:
To find the velocity of point Q, we can use the concept of rolling motion. When a wheel rolls without slipping, the velocity of the center of the wheel is equal to the velocity of any point on the wheel.

Given:
Velocity of point P = 10 m/s

Approach:
To find the velocity of point Q, we can consider the motion of the wheel as a combination of two motions:
1. Translation: The center of the wheel moves in a straight line with a constant velocity. Since the wheel is rolling without slipping, the velocity of the center of the wheel is equal to the velocity of point P.
2. Rotation: The wheel is also rotating about its center. The linear velocity of any point on the wheel due to rotation depends on its distance from the center.

Calculating the Velocity of Point Q:
Since the wheel is rolling without slipping, the velocity of the center of the wheel is equal to the velocity of point P, which is 10 m/s.

The velocity of point Q can be calculated using the formula:
vQ = vP + vQ/P

where vQ is the velocity of point Q,
vP is the velocity of point P,
and vQ/P is the relative velocity of point Q with respect to point P.

Relative Velocity of Point Q with Respect to Point P:
The relative velocity of point Q with respect to point P can be calculated using the formula:
vQ/P = ω × r

where ω is the angular velocity of the wheel and r is the radius of the wheel.

Since the wheel is rolling without slipping, the distance traveled by the wheel in one complete rotation is equal to the circumference of the wheel, which is 2πr.

The time taken for one complete rotation is the period of rotation, which is equal to the time taken for the wheel to travel a distance equal to its circumference.

Therefore, the angular velocity ω can be calculated as:
ω = 2πr / T

where T is the time taken for one complete rotation.

Calculating the Time Taken for One Complete Rotation:
The time taken for one complete rotation can be calculated using the formula:
T = 2π / ω

where ω is the angular velocity of the wheel.

Substituting the Values:
Substituting the values into the equation, we get:
T = 2π / (2πr / T)

Simplifying the equation, we get:
T^2 = r

Calculating the Relative Velocity of Point Q with Respect to Point P:
Substituting the value of T into the equation for ω, we get:
ω = 2πr / T

Substituting the values into the equation for vQ/P, we get:
vQ/P = ω × r

Simplifying the equation, we get:
vQ/P = 2πr / T × r

Substituting the value of T^2, we get:
v