15. Using your derivations in the previous two questions, calculate the coefficient of kinetic triction and the uncertainty Sµk. 16. Report your final value for Hxt8ur:
15. Using your derivations in the previous two questions, calculate the coefficient of kinetic triction and the uncertainty Sµk. 16. Report your final value for Hxt8ur:
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Please answer all parts of problems 15 and 16 only showing all work. Thank You!!!
Transcribed Image Text:The document contains handwritten calculations explaining the derivation of the coefficient of kinetic friction and its uncertainty, using free body diagrams (FBD) and equations. Here is a detailed transcription and explanation:
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### Instructions and Calculations:
**13.** Using the FBD you drew in the prelab, set up both ΣF equations, and use them to derive the expression for the coefficient of kinetic friction. Show all work! You may use additional sheets of paper if necessary (remember ΣF ≠ 0!).
- Start with the equations derived from the FBD:
- ΣF_x = m * a
- ΣF_y = 0
- Resolve forces along axes:
- N = mg cos(θ)
- mgsin(θ) - μ_kN = ma
- Express μ_k in terms of given variables:
- μ_k = (g sin(θ) - a) / (g cos(θ))
- Final Expression: μ_k = tan(θ) - a / g cos(θ)
**14.** You should’ve derived μ_k = tan(θ) - a / g cos(θ). Using propagation of error analysis, derive the expression for the uncertainty in μ_k. Take g to be exact. Show all work!
- Calculate δμ_k using error propagation:
- δμ_k = √( (∂μ_k/∂θ * δθ)^2 + (∂μ_k/∂a * δa)^2 )
- Simplified to: δμ_k = g sin(θ) - a / g cos(θ) * √( (sin(θ) δθ / cos(θ))^2 + (δa / a)^2 + (δg / g)^2 )
**15.** Using your derivations in the previous two questions, calculate the coefficient of kinetic friction and the uncertainty δμ_k.
**16.** Report your final value for μ_k ± δμ_k: __________
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This document guides you through a set of equations, derivations, and error analysis to calculate the coefficient of kinetic friction (μ_k) and its associated uncertainty. The handwritten notes provide step-by-step algebraic manipulations and utilize calculus concepts for deriving the uncertainty formula.
Transcribed Image Text:**Purpose**
To practice with drawing FBDs (Free-Body Diagrams) and working with net force; to learn to work with inclined planes and frictional forces.
**Introduction**
Newton's Laws assert that if a particle is in equilibrium, then the total force on it must vanish, i.e., the vector sum of the applied forces must be equal to zero, \(\sum_i \vec{F_i} = 0\). If the total force is not zero, the particle is not in equilibrium, and then \(\sum_i \vec{F_i} = \vec{ma}\).
The purpose of this experiment is to work with a system that can be in equilibrium, or not in equilibrium (what is the main difference and how can you tell?). We will also practice drawing FBD and working with friction.
**Prelab**
1. Below is a schematic of an inclined plane problem. In the space provided, draw a free-body diagram and label all the forces acting on the box. How can you tell if this box is in equilibrium or not?
*Diagram Description:*
The image shows a box resting on an incline supported by a structure. The incline is positioned at an angle, with the box on its surface. A grid-like structure suggests that the components acting on the box can include gravitational force, normal force, and frictional force.
2. Is there a difference between drawing an FBD for a static case vs. kinetic case? Why or why not?
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