Find the magnitude of the displacement of a point on a wheel initially in contact with the ground when the wheel (of radius 36.2 cm) rolls forward half a revolution.
To find the magnitude of the displacement of a point on a wheel when it rolls forward half a revolution, we need to determine the distance traveled by that point on the wheel.
First, we need to find the circumference of the wheel. The circumference of the wheel can be calculated using the formula: C = 2πr, where r is the radius of the wheel.
Given that the radius of the wheel is 36.2 cm, we can substitute this value into the formula to find the circumference:
C = 2π(36.2) cm
Next, we need to find the distance traveled by the point on the wheel when it completes half a revolution. Since the wheel is rolling forward half a revolution, the point on the wheel is moving along a circular arc halfway around the circumference of the wheel.
To find the distance traveled by the point, we need to calculate half the circumference of the wheel. This can be found by dividing the circumference by 2:
Distance traveled = C/2
Substituting the value of the circumference we calculated earlier, we have:
Distance traveled = (2π(36.2) cm)/2
Now, we can simplify this expression to find the distance traveled by the point:
Distance traveled = π(36.2) cm
Finally, to find the magnitude of the displacement, we use the simple fact that the magnitude of a displacement is equal to the distance traveled. Therefore, the magnitude of the displacement of the point on the wheel is π(36.2) cm.