Question

(b) A plane is flying with a constant speed along a straight line at an angle of 30° with the horizontal.
The weight of the plane is 80, 000 N and its engine provides a thrust of 100, 000 N in the
direction of flight. Two additional forces are exerted on the plane: the lift force perpendicular to
the plane’s wings, and the force due to air resistance opposite to the direction of motion. Draw
the free-body diagram showing all forces on the plane. Determine the lift force and the force due
to air resistance.
2. A bus is moving downhill at a slope of 5° on a rainy day. At the moment when the speed of the bus is
30 km h−1,the driver spots a deer 30 m ahead. He applies the brakes and comes to a stop. The deer is
paralyzed by fear and does not move. Will the bus stop

Answer 1

A plane is flying with a constant speed along a straight line at an angle of $30{}^{\circ}$ with the horizontal. The weight of the plane is 80,000 N and its engine provides a thrust of 100,000 N in the direction of flight. Two additional forces are exerted on the plane: the lift force perpendicular to the plane's wings, and the force due to air resistance opposite to the direction of motion. Draw the free-body diagram showing all forces on the plane. Determine the lift force and the force due to air resistance.

Solution: as you can see on the force diagram above, the plane is affected to such forces as weight force $F_{w} = 80,000$ N, force of the air resistance $F_{R}$ , wings lifting force $F_{L}$ , and the moving thrust force $F_{T} = 100,000$ N. All these forces are in the equilibrium, according to the third Newton's law.

The lifting force balances the part of the weight force, so if we'll compare their projections on the plane vertical axis, they are equal: $\vec{F}_L = \vec{F}_W^{vert}$ , $F_L = F_{w'}\cos\gamma = 80,000\cdot\cos30{}^\circ = 69,282$ N

And the thrust force balances the air resistance force and the rest of the weight force, so that the sum of their projections on the plane horizontal axis are equal: $\vec{F}_T = \vec{F}_W^{hor} + \vec{F}_R$ , $F_T = F_{w'}\sin\gamma + F_R$

$$F _ {R} = F _ {T} - F _ {w ^ {\prime}} \sin \gamma = 1 0 0, 0 0 0 - 8 0, 0 0 0 \cdot \sin 3 0 {}^ {\circ} = 6 0, 0 0 0 N$$

Answer: The lift force of the plain wings is 69,282 N, and the force of the air resistance is 60,000 N.