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 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 m above the platform, and the spring constant is 4 000 N/m. Find (a) the firefighter’s speed immediately 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

Work Done by Friction

Friction is a non-conservative force that dissipates mechanical energy, typically transforming some of it into thermal energy. The work done by friction is calculated by multiplying the frictional force by the distance over which it acts and is subtracted from the system's total mechanical energy, affecting the final kinetic or stored potential energy.

Conservation of Mechanical Energy

This principle involves the understanding that the total mechanical energy of a system—comprising kinetic and potential energies—remains constant when only conservative forces (like gravity and spring forces) act. In cases where non-conservative forces like friction are present, the energy lost through work done by these forces must be accounted for in the energy conservation equation.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its height within a gravitational field. It is typically expressed as mgh, where m is mass, g is the gravitational acceleration, and h is the height relative to a chosen reference level. This energy is converted into kinetic energy as the object descends.

Spring Potential Energy

Spring potential energy is the energy stored in a spring when it is compressed or extended from its equilibrium position. It is quantified by the formula (1/2)kx², where k is the spring constant and x is the displacement from equilibrium. This stored energy can then be converted back into kinetic energy as the spring returns to its natural length.

Transcript

00:01 In this question, we have this situation.

00:05 So a firefighter is laying down from a pole for four meters.

00:12 And then at the bottom, there's a platform.

00:17 And then this platform is supported by spring.

00:23 So in part a, we want to find the firefighter speed immediately before she collides with the platform.

00:29 Okay, so along the pole, there is a friction force of 300 newtons that resist the motion.

00:42 Okay, so for the free body diagram of the firefighter, we have the weight and we also have the friction acting on the firefighter.

00:53 So to find the speed, i'll be using work energy theorem.

01:06 Which says that the network done is the change in kinetic energy of the object.

01:14 So the network done for this firefighter would be m g minus friction times h and this is equal to half mv square.

01:27 And then the speed will just be 2 over m g minus f h squared okay putting in the numbers two times four which is the height and then 9 .8 minus 300 divided by 75 okay and then this is 6 .81 meters per second okay so this is the answer for part a then for part b we want to wants to find the maximum distance the spring is compressed.

02:24 Okay, so there is an elastic collision with the platform or perfectly inelastic collision with the platform.

02:45 So conservation of momentum equation would give us the firefighter times the mess of the firefighter times speed, which is this, 6 .81 is equal to the firefighter plus the platform, times the final speed.

03:15 So we get v -prime and then this is our world, actually final momentum, okay? i'm just going to leave it as that.

03:28 Then the spring is compressed after that...

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