Final answer:
The slowest point on a rolling wheel is the point in contact with the ground, traveling at 0 m/s. The fastest point is at the top of the wheel. For a wheel with a rotational speed of 5π rad/sec and a 25-inch diameter, the top point travels at approximately 4.977 m/s.
Step-by-step explanation:
Understanding Point Speeds on a Rolling Wheel
When considering a wheel of diameter 25 inches rolling forward, the points on the wheel will have different speeds relative to the ground due to the wheel's rotational motion. Here's the breakdown of how the speeds vary:
- The point on the wheel that is traveling the slowest relative to the ground at any given time t is the point in contact with the ground. The speed of this point relative to the ground is zero because it is the instantaneous point of rotation.
- The point that is traveling the fastest relative to the ground at any given time t is the point at the top of the wheel. This is because it has the maximum linear velocity due to its maximum radius from the point of rotation.
- The speed of the fastest point on the wheel, when the rotational speed is 5π rad/sec, can be calculated using the formula v = rw, where r is the radius of the wheel and w is the angular velocity. With a 25-inch diameter, the radius is 12.5 inches, which is about 0.3175 meters. Thus, the fastest point speed v is v = 0.3175 meters × 5π rad/sec = 5π × 0.3175 m/s, which equals approximately 4.977 m/s.