Spider-Man Spider-Man holds the bottom of an elevator with...

Spider-Man Spider-Man holds the bottom of an elevator with one hand. With his other hand, he holds a spider cord attached to a $50-\mathrm{kg}$ box of explosives at the bottom of the cord. Determine the force that the cord exerts on the box if (a) the elevator is at rest; (b) the elevator accelerates up at $2.0 \mathrm{m} / \mathrm{s}^{2} ;(\mathrm{c})$ the upward-moving elevator's speed decreases at a rate of $2.0 \mathrm{m} / \mathrm{s}^{2} ;$ and ( $\mathrm{d}$ ) the elevator falls freely.

Spider-Man Spider-Man holds the bottom of an elevator with one hand. With his other hand, he holds a spider cord attached to a $50-\mathrm{kg}$ box of explosives at the bottom of the
cord. Determine the force that the cord exerts on the box if (a) the elevator is at rest; (b) the elevator accelerates up at $2.0 \mathrm{m} / \mathrm{s}^{2} ;(\mathrm{c})$ the upward-moving elevator's speed decreases at a rate of $2.0 \mathrm{m} / \mathrm{s}^{2} ;$ and ( $\mathrm{d}$ ) the elevator falls freely.

Key Concepts

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. This fundamental principle is used to analyze the forces acting on objects in both inertial and non-inertial reference frames, allowing us to determine unknown forces such as tension in a cord or the apparent weight of an object under various acceleration conditions.

Non-Inertial Reference Frames

Non-inertial reference frames are accelerating frames of reference where additional apparent forces, sometimes referred to as fictitious forces or pseudo forces, must be introduced to account for the acceleration of the frame. When analyzing problems in such frames (like an accelerating elevator), the effective force on an object is modified by the acceleration of the frame, changing its apparent weight and the required force to support or accelerate it.

Apparent Weight

Apparent weight is the perceived weight of an object when it is in an accelerating system, which can differ from its real gravitational weight. In scenarios where the frame of reference accelerates upward or downward, the apparent weight is calculated by combining gravitational acceleration with the acceleration of the frame, thus altering the tension in supporting structures like cords or ropes.

Free Fall

Free fall describes the condition where the only force acting on an object is gravity. In free-fall situations, the object experiences weightlessness because the acceleration due to gravity is not countered by any supporting force, leading to a zero apparent weight and, consequently, zero tension in any connection that would normally support its weight.

Transcript

00:01 We have this 50 -kilogram box of explosives attached to the bottom of an elevator with this cord.

00:07 And we want to determine the force the cord exerts on the box of explosives under four different circumstances.

00:13 When the elevator is not moving, if it's accelerating upwards at 2 .0 meters per second squared, accelerating downwards at 2 .0 meters per second squared, and in free fall.

00:26 So if we look at the case where the elevator is not moving, and we look at the forces on this box.

00:48 We can see that we have a downward force due to gravity.

00:52 That's going to be equal to the mass times the acceleration due to gravity, which is g.

00:58 And then that's going to be 50 kilograms times 9 .8 meters per second squared is 490 newtons.

01:14 And we also have an upward force acting on.

01:16 On this box.

01:17 We'll call it f sub c for the force of the cord.

01:21 So when the elevator is not moving, it's not accelerating, and therefore this box, which is attached to the elevator and is part of the system, is also not accelerating.

01:32 So the net force must be zero since there is no acceleration.

01:42 And then the net force is also the sum of all of the forces acting on the object.

01:47 So here the object is the box.

01:50 And we have two forces, the force due to the cord, which we're trying to look for, and the force due to gravity, 490 newtons.

01:57 So we'll sign upwards as the positive direction.

02:01 So the force of the cord is going to be positive.

02:04 And then we also want to add the force due to gravity, but since that's downward, that's negative, and when you add a negative, you subtract.

02:12 So we subtract 490 newtons, the force due to gravity.

02:18 So we can add 490 newtons to both sides.

02:27 And that gives us 490 newtons equals the force of the cord.

02:35 So when the elevator is not moving, the cord exerts a force of 490 newtons, and that makes sense because then it can cancel out the force due to gravity.

02:46 So the next case, the elevator is accelerating.

02:51 Meters per second squared upwards.

02:55 So we'll go back to our force diagram here...

Please give Ace some feedback