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Chapter 7: Problem 56

\(\bullet\) A fun-loving 11.4 \(\mathrm{kg}\) otter slides up a hill and then back down to the same place. If she starts up at 5.75 \(\mathrm{m} / \mathrm{s}\) and returns at 3.75 \(\mathrm{m} / \mathrm{s}\) , how much mechanical energy did she lose on the hill, and what happened to that energy?

Short Answer

Expert verified
The otter lost 108.48 J of mechanical energy, converted mainly to thermal energy due to friction.

Step by step solution

01

Identify the Given Values

We're given that the otter has a mass of 11.4 kg. Her initial speed, when sliding up, is 5.75 m/s, and her final speed, when she returns to the starting point, is 3.75 m/s. Our task is to find out how much mechanical energy was lost due to the slide.
02

Calculate Initial Kinetic Energy

The initial kinetic energy (KE_initial) when she starts sliding up can be calculated using the formula:\[KE_{initial} = \frac{1}{2} m v_{initial}^2\]Substituting the known values:\[KE_{initial} = \frac{1}{2} \times 11.4 \, \text{kg} \times (5.75 \, \text{m/s})^2 = 188.92 \, \text{Joules}\]
03

Calculate Final Kinetic Energy

The final kinetic energy (KE_final) when she returns is calculated as follows:\[KE_{final} = \frac{1}{2} m v_{final}^2\]Substitute the given values:\[KE_{final} = \frac{1}{2} \times 11.4 \, \text{kg} \times (3.75 \, \text{m/s})^2 = 80.44 \, \text{Joules}\]
04

Determine Mechanical Energy Lost

The mechanical energy lost is the difference between the initial and final kinetic energies:\[\text{Energy Lost} = KE_{initial} - KE_{final}\]Calculating:\[\text{Energy Lost} = 188.92 \, \text{Joules} - 80.44 \, \text{Joules} = 108.48 \, \text{Joules}\]
05

Analyze What Happened to the Lost Energy

The lost mechanical energy, amounting to 108.48 Joules, was transformed into other forms of energy. Most likely, this includes thermal energy due to friction between the otter and the hill surface, and possibly sound energy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy
Kinetic energy is a form of energy that an object possesses due to its motion. It depends on two main factors: the object's mass and its velocity. The formula to calculate kinetic energy is:
  • \[ KE = \frac{1}{2} m v^2 \]
Where \( KE \) is the kinetic energy, \( m \) is the mass, and \( v \) is the velocity of the object.
For instance, when the otter in the problem slides up the hill, she has an initial kinetic energy calculated using her speed of 5.75 m/s. Similarly, her kinetic energy can be recalculated when she returns to the bottom at a different speed of 3.75 m/s.
Observing these changes in energy is crucial to understanding how energy is conserved or transformed during motion.
Energy Transformation
Energy transformation is the process where energy changes from one form to another. Here, it's essential to understand that while the total energy of the system remains constant in an ideal situation (as per the principle of conservation of energy), the form of energy can change.
In the case of the otter sliding on the hill, her potential energy increases as she moves up the hill, and her kinetic energy decreases. Conversely, when sliding back down, potential energy decreases while kinetic energy increases.
However, in real-life scenarios, some of the kinetic energy might not fully convert back, leading to energy transformations into non-mechanical forms like thermal energy due to friction, which comes into play here.
Energy Loss Due to Friction
Energy loss due to friction is a common phenomenon where mechanical energy is converted into other forms like heat. Friction is a force that opposes motion between two surfaces in contact, causing some of the kinetic energy to dissipate.
In the otter's case, as she slides up and down the hill, friction between her and the surface of the hill converts a portion of her kinetic energy into heat. This is why, when she returns to the starting point, her speed is lower than when she began.
  • The amount of energy loss can often be quantified, as in the exercise where the otter lost 108.48 Joules due to friction.
Understanding energy loss is important for grasping real-life applications of mechanical energy conservation and transformations.

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Most popular questions from this chapter

\(\bullet\) \(\bullet\) A spring is 17.0 \(\mathrm{cm}\) long when it is lying on a table. One end is then attached to a hook and the other end is pulled by a force that increases to 25.0 \(\mathrm{N}\) , causing the spring to stretch to a length of 19.2 \(\mathrm{cm} .\) (a) What is the force constant of this spring? (b) How much work was required to stretch the spring from 17.0 \(\mathrm{cm}\) to 19.2 \(\mathrm{cm}\) (c) How long will the spring be if the 25 \(\mathrm{N}\) force is replaced by a 50 \(\mathrm{N}\) force?

\(\bullet\) \(\bullet\) \(\mathrm{A} 25 \mathrm{kg}\) child plays on a swing having support ropes that are 2.20 \(\mathrm{m}\) long. A friend pulls her back until the ropes are \(42^{\circ}\) from the vertical and releases her from rest. (a) What is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing? (b) How fast will she be moving at the bottom of the swing? (c) How much work does the tension in the ropes do as the child swings from the initial position to the bottom?

\(\bullet\) Bumper guards. You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1200 \(\mathrm{kg}\) car moving at 0.65 \(\mathrm{m} / \mathrm{s}\) is to compress the spring no more than 7.0 \(\mathrm{cm}\) before stopping. (a) What should be the force constant of the spring, and what is the maximum amount of energy that gets stored in it? (b) If the springs that are actually delivered have the proper force constant but can become compressed by only \(5.0 \mathrm{cm},\) what is the maximum speed of the given car for which they will provide adequate protection?

\(\bullet\) \(\bullet\) Automobile accident analysis. In an auto accident, a car hit a pedestrian and the driver then slammed on the brakes to stop the car. During the subsequent trial, the driver's lawyer claimed that the driver was obeying the posted 35 mph speed limit, but that the limit was too high to enable him to see and react to the pedestrian in time. You have been called as the state's expert witness. In your investigation of the accident site, you make the following measurements: The skid marks made while the brakes were applied were 280 ft long, and the tread on the tires produced a coefficient of kinetic friction of 0.30 with the road. (a) In your testimony in court, will you say that the driver was obeying the posted speed limit? You must be able to back up your answer with clear numerical reasoning during cross-examination. (b) If the driver's speeding ticket is \(\$ 10\) for each mile per hour he was driving above the posted speed limit, would he have to pay a ticket, and if so, how much would it be?

\(\bullet\) A 12.0 g plastic ball is dropped from a height of 2.50 \(\mathrm{m}\) and is moving at 3.20 \(\mathrm{m} / \mathrm{s}\) just before it hits the floor. How much mechanical energy was lost during the ball's fall?
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