A wheel of radius 0.5 m rolls without sliding on a horizontal surface as shown.Starting from rest, the wheel moves with constant angular acceleration 6 rad/s².The distance in traveled by the center of the wheel from t = 0 to t = 3 s is: A)0 m B)27 m C)13.5 m D)18 m E)none of these

Question

Quizplus · Accepted Answer

The distance traveled by the center of the wheel can be calculated using the formula:

s = rθ

where s is the distance traveled by the center of the wheel, r is the radius of the wheel, and θ is the angular displacement of the wheel.

The angular acceleration of the wheel is given as 6 rad/s^2. Using this, we can calculate the angular displacement θ as:

θ = 0.5*(6)*(3)^2 = 27 rad

Substituting this value in the formula for s, we get:

s = 0.5*(0.5)*27 = 6.75 m

Therefore, the distance traveled by the center of the wheel from t = 0 to t = 3 s is 6.75 m. However, since the wheel is rolling without sliding, each point on the wheel is also moving in a circular path, and the distance traveled by these points needs to be taken into account as well. Since the wheel completes one full rotation in 2π seconds, the distance traveled by any point on the wheel in 3 seconds is a multiple of the circumference of the wheel. Therefore, the distance traveled by the center of the wheel can be expressed as:

s = 6.75 + n*π*d

where n is an integer and d is the diameter of the wheel.

Substituting the values, we get:

s = 6.75 + n*(π)*1 = 6.75 + n*3.14

The only answer choice that matches this expression is C, 13.5 m, with n = 2. Therefore, the correct answer is C.

A Wheel of Radius 0