Your spaceship has been designed with a large rotating wheel to give an impression of gravity. The radius of the wheel is \(R=50 \mathrm{~m}\). (a) How many rotation per minutes must the wheel execute for the acceleration at the outer end of the wheel to correspond to the acceleration of gravity at the Earth, \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) ? (b) What is the difference in acceleration of your feet and your head if you are standing with your feet at the outer end of the rotating wheel? You can assume that you are approximately \(2 \mathrm{~m}\) high.

Rotational motion refers to the motion of an object around a central point or axis. Imagine a spinning Ferris wheel—each seat moves in a circular path. In your exercise, the spaceship's rotating wheel simulates gravity. Here, the wheel's rotation creates a centripetal force that pushes you outward, simulating the feeling of gravity. When an object moves in a circle, its speed is called its angular velocity, and the time it takes to make one full circle is called its period. To feel like you're experiencing Earth-like gravity, the wheel's centripetal acceleration should match the gravitational acceleration of 9.8 m/s².

Gravitational acceleration is the acceleration of an object due to the force of gravity. On Earth, this acceleration is approximately 9.8 m/s². This value means that, in the absence of air resistance, an object in free fall will speed up by about 9.8 meters per second every second. In your exercise, the rotating spaceship wheel needs to mimic this acceleration. By setting the centripetal acceleration equal to the gravitational acceleration, we can ensure that the simulated 'gravity' feels the same as Earth’s gravity. That’s why we use the centripetal acceleration formula: \( a_c = \frac{v^2}{R} \), where \(a_c\) is the centripetal acceleration, \(v\) is the linear velocity, and \(R\) is the radius of the circle. Setting \( a_c = g \) allows us to solve for the needed velocity.

Centripetal force is the force required to keep an object moving in a circular path. It acts towards the center of the circle and is crucial for rotational motion. Without this force, an object would move in a straight line due to inertia. For the spaceship wheel, the centripetal force is what keeps you 'stuck' to the outer edge. The larger the radius and the faster the rotation, the greater the required centripetal force. This force results from the centripetal acceleration: \( a_c = \frac{v^2}{R} \). For our spaceship wheel to create a force that feels like Earth’s gravity, its centripetal acceleration at the edge must equal 9.8 m/s². This alignment ensures that you'd experience the same force pushing you against the wheel as you would standing on Earth's surface.

Angular velocity measures how quickly an object rotates or revolves relative to another point. It’s usually expressed in radians per second (rad/s). In the exercise, we calculate the angular velocity needed to experience Earth-like gravity on the spaceship. The relationship between linear velocity \(v\) and angular velocity \(\omega\) is given by the formula: \( v = \omega R \). By solving for \( \omega \), we find the rate of rotation needed for the spaceship wheel. After computing \( \omega \) in rad/s, we convert it to rotations per minute (rpm) to find how many rotations per minute the wheel must make. For instance, if \( \omega = 0.4428 \) rad/s, converting this to rotations per minute gives approximately 4.23 rpm. This rotation rate ensures the centripetal acceleration at the wheel's edge equals Earth’s gravitational acceleration, emulating gravity for the spaceship occupants.