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0. Math Review31m
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1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

7. Friction, Inclines, Systems

Kinetic Friction

Physics

7. Friction, Inclines, Systems

Kinetic Friction

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7:26 minutes

Problem 78c

Textbook Question

A 1.0-cm-diameter, 2.0 g marble is shot horizontally into a tank of 20°C olive oil at 10 cm/s. How far in cm will it travel before stopping?

Verified step by step guidance

Determine the forces acting on the marble as it moves through the olive oil. The primary force opposing its motion is the drag force, which can be expressed as $ F_d = \frac{1}{2} C_d \rho A v^2 $, where $ C_d $ is the drag coefficient, $ \rho $ is the density of the fluid (olive oil), $ A $ is the cross-sectional area of the marble, and $ v $ is its velocity.

Calculate the cross-sectional area $ A $ of the marble using the formula for the area of a circle: $ A = \pi r^2 $, where $ r $ is the radius of the marble. The radius can be found by dividing the diameter (1.0 cm) by 2.

Write the equation of motion for the marble. The net force acting on the marble is $ F = ma $, where $ m $ is the mass of the marble and $ a $ is its acceleration. Since the drag force is the only force acting in the horizontal direction, $ ma = -F_d $. Substitute $ F_d $ into this equation to express the acceleration in terms of velocity: $ a = -\frac{1}{2m} C_d \rho A v^2 $.

Recognize that the velocity $ v $ decreases over time due to the drag force. Use the relationship between acceleration and velocity: $ a = \frac{dv}{dt} $. Substitute $ a $ from the previous step into this equation to obtain $ \frac{dv}{dt} = -\frac{1}{2m} C_d \rho A v^2 $. Rearrange to separate variables: $ \frac{dv}{v^2} = -\frac{1}{2m} C_d \rho A dt $.

Integrate both sides of the equation to find the distance traveled by the marble before it stops. The left-hand side integrates with respect to velocity from $ v = 10 \ \text{cm/s} $ to $ v = 0 $, and the right-hand side integrates with respect to time. After solving for time, use the relationship $ v = \frac{dx}{dt} $ to find the total distance traveled by the marble.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Drag Force

The drag force is the resistance experienced by an object moving through a fluid, such as olive oil. It depends on the object's velocity, the fluid's density, and the object's cross-sectional area. The drag force acts opposite to the direction of motion and increases with speed, ultimately affecting how far the marble will travel before stopping.

Terminal Velocity

Terminal velocity is the constant speed an object reaches when the drag force equals the gravitational force acting on it, resulting in no net acceleration. In this scenario, the marble will decelerate due to drag until it reaches a point where the forces balance, which is crucial for determining how far it will travel in the olive oil.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this problem, these equations can be used to calculate the distance the marble travels before coming to a stop, taking into account the deceleration caused by the drag force.

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