A 75.0 -kg firefighter slides down a pole while a constant...

A 75.0 -kg firefighter slides down a pole while a constant friction force of 300 $\mathrm{N}$ retards her motion. A horizontal 20.0 -kg platform is supported by a spring at the bottom of the pole to cushion the fall. The firefighter starts from rest 4.00 $\mathrm{m}$ above the platform, and the spring constant is 4000 $\mathrm{N} / \mathrm{m}$ . Find (a) the firefighter's speed just before she collides with the platform and (b) the maximum distance the spring is compressed. Assume the friction force acts during the entire motion.

A 75.0 -kg firefighter slides down a pole while a constant friction force of 300 $\mathrm{N}$ retards her motion. A horizontal 20.0 -kg platform is supported by a spring at the bottom of the pole to cushion the fall. The firefighter starts from rest 4.00 $\mathrm{m}$ above the platform, and the spring constant is
4000 $\mathrm{N} / \mathrm{m}$ . Find (a) the firefighter's speed just before she collides with the platform and (b) the maximum distance the spring is compressed. Assume the friction force acts during the entire motion.

Key Concepts

Spring Potential Energy and Hooke’s Law

This concept involves the energy stored in a spring when it is compressed or stretched. According to Hooke’s Law, the force exerted by a spring is proportional to its displacement from equilibrium, and the spring potential energy is given by one-half the spring constant times the square of the displacement. This energy plays a key role in cushioning impacts and absorbing kinetic energy.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. In energy conversion problems, it usually represents the energy gained as an object moves faster, often at the expense of potential energy. This concept is central when analyzing the speed of an object just before impact.

Work Done by Non-Conservative Forces

Non-conservative forces, such as friction, perform work that dissipates mechanical energy from the system. In problems where such forces are present, the work done by these forces is subtracted from the initial total energy, leading to a lower final mechanical energy than would be predicted by conservative forces alone.

Conservation of Energy

This concept involves the idea that the total energy in a closed system remains constant, though it can transform between different forms such as kinetic, potential, and elastic energy. When non-conservative forces (like friction) are present, they do work that converts mechanical energy into other forms (like thermal energy), so energy conservation must account for these losses.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is calculated as the product of the object's mass, the gravitational acceleration, and its height above a reference point, and it is converted to kinetic energy as the object falls.

Transcript

00:01 Okay, so here you have a firefighter that's sliding down this pore here.

00:09 You're told that the poor in total is 4 meters long with this firefighter with about 75 kilograms.

00:22 And you're told that it's a 300 newton frictional force acting on this system.

00:31 And you're told that he lands on a platform that was that is 20 kilograms and that platform is supported by a spring that it has a spring constant k of 4 ,000 moutons per meter here so essentially you want first you want to calculate the velocity that it hits a platform with and then after that you want to see what kind of compression induces with that velocity so the first thing is first.

01:10 We want to find out how much speed this guy has when we get to the bottom.

01:17 So essentially this is a second low problem.

01:19 So you have the force of gravity minus this friction of force here.

01:32 This is so this is summing up all the forces that should give you the resultant force, which is mass -time acceleration.

01:39 So these are the two forces acting on this person as they slide down.

01:44 So you've noticed that this we can calculate this is just 75 times 9 .8 minus 300 and then you divide that by the mass of person which is just 75.

02:04 So once you do this you actually find the acceleration and the acceleration in this case is about 5 .8.

02:14 Meters for a square second and then what you want to calculate now is so once you have that now we want to find the velocity once they hit the bottom so we know that we don't we don't have anything in terms of time here so we want to use kinemite equations that don't involve time so the best one to use is that one that says vbano squared is equal to v initial squared plus plus 2a times of change in distance.

02:52 And in which case the change in distance here is just y.

02:55 So we know that the v initial here is zero because we're told that it starts from rest.

03:02 We assume it starts from rest if we are not told that.

03:05 So this a is what we just calculated here.

03:08 So that goes here and then this delta y is the four meters.

03:12 So we know that the fine velocity is going to be the square root of 2.

03:21 Two times the acceleration 5 .8 times the distance which is four and that should give us something like 6 .81 meters per second.

03:42 Then the more complicated about this problem is actually figuring out how much this thing drops back.

03:49 So we can't use kinematics in this in this scenario first, we have several different things going on...

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