The drawing shows a plane diving toward the ground and then...

The drawing shows a plane diving toward the ground and then climbing back upward. During each of these motions, the lift force $\overrightarrow{\mathbf{L}}$ acts perpendicular to the displacement $\overrightarrow{\mathbf{s}},$ which has the same magnitude, $1.7 \times 10^{3} \mathrm{m},$ in each case. The engines of the plane exert a thrust $\mathbf{T},$ which points in the direction of the displacement and has the same magnitude during the dive and the climb. The weight $\overrightarrow{\mathbf{W}}$ of the plane has a magnitude of $5.9 \times 10^{4} \mathrm{N} .$ In both motions, net work is performed due to the combined action of the forces $\overrightarrow{\mathbf{L}}, \overrightarrow{\mathbf{T}},$ and $\overrightarrow{\mathbf{W}}$. (a) Is more net work done during the dive or the climb? Explain. (b) Find the difference between the net work done during the dive and the climb.

The drawing shows a plane diving toward the ground and then climbing back upward. During each of these motions, the lift force $\overrightarrow{\mathbf{L}}$ acts perpendicular to the displacement $\overrightarrow{\mathbf{s}},$ which has the same magnitude, $1.7 \times 10^{3} \mathrm{m},$ in each case. The engines of the plane exert a thrust $\mathbf{T},$ which points in the direction of the displacement and has the same magnitude during the dive and the climb. The weight $\overrightarrow{\mathbf{W}}$ of the plane has a magnitude of $5.9 \times 10^{4} \mathrm{N} .$ In both motions, net work is performed due to the combined action of the forces $\overrightarrow{\mathbf{L}}, \overrightarrow{\mathbf{T}},$ and $\overrightarrow{\mathbf{W}}$.
(a) Is more net work done during the dive or the climb? Explain. (b) Find the difference between the net work done during the dive and the climb.

Key Concepts

Work

Work is the energy transferred when a force causes a displacement and is defined mathematically as the dot product of the force and displacement vectors. This means that only the component of the force parallel to the displacement does work. A force perpendicular to the displacement, such as the lift in this scenario, does no work.

Dot Product

The dot product is a mathematical operation that measures the extent to which two vectors point in the same direction. In the context of work, it multiplies the magnitude of the force by the magnitude of the displacement and by the cosine of the angle between them, which is why forces perpendicular to displacement contribute no work.

Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. This principle connects the concept of work with the dynamics of motion, allowing us to analyze how different forces (with different orientations) contribute differently to the energy of the system.

Gravitational Work

Gravitational force (weight) always points downwards, so its work depends on the direction of displacement. When the plane dives (moving downward), weight contributes positive work, while during the climb (moving upward), weight contributes negative work. This difference in gravitational work is key to understanding why the net work differs between the dive and the climb.

Force Decomposition

In analyzing forces acting on an object, it is essential to decompose them into components parallel and perpendicular to the direction of displacement. Only the parallel components perform work, as is illustrated by the thrust (parallel) and lift (perpendicular) forces in this problem. This decomposition helps in assessing the overall contribution of various forces to the net work done on the object.

Transcript

Please give Ace some feedback