91Ó°ÊÓ

Work and energy are fundamental concepts in physics. Work is defined as the transfer of energy when a force is applied to an object over a distance. It's given by the formula: \[ W = F \times d \times \text{cos} \theta \], where \(F\) is the force, \(d\) is the distance, and \(\theta\) is the angle between the force and the direction of movement. In the exercise, the helicopter applies a force to lift the astronaut, performing positive work since the force and displacement are in the same direction. On the other hand, gravitational force does negative work because it acts in the opposite direction to the displacement. Energy, in this case, can be in various forms, such as kinetic energy, potential energy, or mechanical energy. Understanding how these energies transform is key to solving many physical problems.

Newton's second law is crucial when dealing with forces and motion. The law states that the net force on an object is equal to its mass times its acceleration: \(F_{net} = ma\). This relationship helps us find the net force acting on the astronaut when the helicopter lifts her. Given the mass \(m = 72 \text{ kg}\) and the acceleration \(a = \frac{g}{10}\), we can calculate the net force as \(F_{net} = 72 \text{ kg} \times \frac{9.8 \text{ m/s}^2}{10} = 70.56 \text{ N}\). This net force adds to the gravitational force acting downward to give the total force exerted by the helicopter. Proper application of Newton's second law is vital in solving real-world problems effectively.

Gravitational force plays a key role in many mechanics problems. This force is given by the equation \(F_g = mg\), where \(m\) is the mass and \(g\) is the acceleration due to gravity (9.8 \(\text{ m/s}^2\) on Earth). In the given problem, the gravitational force on the astronaut is \(F_g = 72 \text{ kg} \times 9.8 \text{ m/s}^2 = 705.6 \text{ N}\). This downward force must be overcome by the helicopter to lift the astronaut. While calculating work done by gravitational force, we use the formula \(W_g = F_g \times d \times \text{cos}\theta\), and since the force acts in the opposite direction to the lift, the work done is negative, explaining the subtraction in total work calculations. Understanding gravitational force is essential for analyzing object motion under Earth's gravity.

Kinetic energy is the energy of motion and is given by the formula \(KE = \frac{1}{2} mv^2\). It tells us how much energy an object has due to its motion. In our problem, once we know the net work done on the astronaut, we can calculate her kinetic energy, indicating how fast she's moving just before reaching the helicopter. We found the net work as \(1058.4 \text{ J}\), which equals her kinetic energy upon lifting. Solving \(KE = \frac{1}{2} \times 72 \times v^2\) for \(v\), we find her speed. This relationship between work and kinetic energy reflects the Work-Energy Principle, showing the energy transformation as forces act on an object, moving it through a distance.

Acceleration is the rate of change of velocity and is a vector quantity, meaning it has both magnitude and direction. It is calculated by \(a = \frac{F_{net}}{m}\). In the exercise, the astronaut's acceleration \(a = \frac{g}{10}\) was given, simplifying calculations. When dealing with problems involving varying accelerations, it's often necessary to break down forces and motions into components to understand better how and why objects move. Acceleration directly affects how quickly the astronaut gains speed as she's lifted, highlighting how understanding the forces and motion in play leads to solving the problems in mechanics.