Question

What are \((a)\) the angular speed, \((b)\) the radial acceleration, and \((c)\) the tangential acceleration of a spaceship negotiating a circular turn of radius \(3220 \mathrm{~km}\) at a constant speed of \(28,700 \mathrm{~km} / \mathrm{h} ?\)

Short answer

The angular speed is approximately \(2.47 \times 10^{-4} s^{-1}\), the radial acceleration is approximately \(0.230 m s^{-2}\), and the tangential acceleration is \(0 m s^{-2}\).

Step-by-step solution

Conversion of Units

In order to work with standard units of measurements, convert the given speed from kilometres per hour (km/h) to metres per second (m/s) and the radius from kilometres (km) to metres (m). Speed in m/s = \( \frac{28700 km/h}{3.6} \). Radius in m = \( 3220 km * 1000 \).

Calculation of Angular Speed

Angular speed (\( \omega \)) is given by the formula \( \omega = \frac{v}{r} \). Substitute the values obtained in Step 1 into this formula to get the angular speed.

Calculation of Radial Acceleration

Radial acceleration (\( a_r \)) is given by the formula \( a_r = \frac{v^2}{r} \). Substitute the values obtained in Step 1 into this formula to get the radial acceleration.

Determining Tangential Acceleration

The tangential acceleration of an object moving with a constant speed in a circular path is zero. Therefore, tangential acceleration (\( a_t \)) = 0.

Key concepts

Angular Speed

Understanding angular speed is crucial when studying objects in circular motion, such as a spaceship navigating a circular path. Angular speed, denoted as \( \omega \) (omega), quantifies how fast an object rotates or revolves relative to the center of the circular path. It is defined as the angle covered per unit time and is usually measured in radians per second (rad/s).To calculate angular speed, we use the formula \( \omega = \frac{v}{r} \) where \( v \) represents the linear (tangential) speed, and \( r \) is the radius of the circular path. In essence, angular speed tells us how quickly the spaceship is changing its position along the circular orbit.

Radial Acceleration

Radial acceleration, also known as centripetal acceleration, occurs when an object is moving along a curved path, such as a circle. This acceleration is always directed toward the center of the circle and is responsible for changing the direction of the object's velocity. The numerical value of radial acceleration is given by the formula \( a_r = \frac{v^2}{r} \) where \( v \) is the linear speed and \( r \) is the radius of the circular path.

For our example of the spacecraft, you can think of radial acceleration as the force that keeps it from flying off into space and instead guides it to follow the circular trajectory. Despite moving at a constant speed, this change in direction means the spaceship is indeed accelerating towards the center, showcasing the non-linear nature of circular motion.

Tangential Acceleration

Tangential acceleration is the rate at which the linear speed of an object moving along a circular path changes. It acts perpendicular to radial acceleration and is parallel to the tangential velocity of the object. It can be calculated when there's an angular acceleration involved. However, if an object moves with a constant speed in its circular path, like the spaceship in our exercise, the tangential acceleration is zero, \( a_t = 0 \) m/s².

This means that while the spaceship is changing its direction continuously, it is not speeding up or slowing down along the path of its orbit. This point is particularly interesting in physics since it allows us to think about acceleration not just as a change in speed, but also as a change in direction.

Unit Conversion

Unit conversion is a fundamental aspect of solving physics problems, ensuring all values are in compatible units before performing calculations. In the solution steps you've read, the spaceship's speed was originally given in kilometers per hour (km/h) and the radius in kilometers (km). However, to work with standard units for calculating angular speed and acceleration, we converted these values to meters per second (m/s) for speed and meters (m) for radius.

To convert from km/h to m/s, we divide the speed by 3.6 (since one hour has 3600 seconds and one kilometer has 1000 meters). Similarly, we multiplied the radius in kilometers by 1000 to get the radius in meters. Without proper unit conversion, our calculations would yield incorrect results, potentially leading to significant misunderstandings of the physical concepts at play.