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What is the role of damping in oscillatory systems?
May 19, 2024
What is the role of damping in oscillatory systems?
Solution
a
Damping in oscillatory systems refers to the presence of a force that opposes the motion of the oscillating object. This force is typically proportional to the velocity of the object and acts in the opposite direction
b
The damping force can be mathematically expressed as: Fd=−bvF_d = -b v where bb is the damping coefficient and vv is the velocity
c
Damping affects the amplitude of oscillations, causing them to decrease over time. In a damped harmonic oscillator, the equation of motion is given by: md2xdt2+bdxdt+kx=0m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 where mm is the mass, bb is the damping coefficient, kk is the spring constant, and xx is the displacement
d
Depending on the value of the damping coefficient bb, the system can be underdamped, critically damped, or overdamped. In an underdamped system, the object oscillates with a gradually decreasing amplitude. In a critically damped system, the object returns to equilibrium as quickly as possible without oscillating. In an overdamped system, the object returns to equilibrium without oscillating but more slowly than in the critically damped case
Answer
Damping in oscillatory systems introduces a force that opposes the motion, reducing the amplitude of oscillations over time. The nature of the damping (underdamped, critically damped, or overdamped) depends on the damping coefficient.
Key Concept
Damping in oscillatory systems: The damping force is proportional to the velocity and acts in the opposite direction, affecting the amplitude and nature of oscillations.
Explanation
Damping reduces the amplitude of oscillations over time and can result in different types of motion (underdamped, critically damped, or overdamped) depending on the damping coefficient.
Calculate the uncertainty of the refractive index of the given prism using the law of uncertainty propagation. Note obtained result along with the uncertainty in the correct format.
Solution
a
Identify the variables involved in the calculation of the refractive index and their uncertainties. Let's denote the refractive index as nn, and the variables involved as x1,x2,…,xnx_1, x_2, \ldots, x_n with their respective uncertainties Δx1,Δx2,…,Δxn\Delta x_1, \Delta x_2, \ldots, \Delta x_n
b
Use the law of uncertainty propagation to calculate the uncertainty in the refractive index. The general formula for uncertainty propagation is given by: Δn=(∂n∂x1Δx1)2+(∂n∂x2Δx2)2+…+(∂n∂xnΔxn)2 \Delta n = \sqrt{\left(\frac{\partial n}{\partial x_1} \Delta x_1\right)^2 + \left(\frac{\partial n}{\partial x_2} \Delta x_2\right)^2 + \ldots + \left(\frac{\partial n}{\partial x_n} \Delta x_n\right)^2} where ∂n∂xi\frac{\partial n}{\partial x_i} represents the partial derivative of nn with respect to xix_i
c
Calculate the partial derivatives of the refractive index with respect to each variable. For example, if the refractive index nn is a function of angle of incidence θi\theta_i and angle of refraction θr\theta_r, then: ∂n∂θi=∂∂θi(sin⁡θisin⁡θr) \frac{\partial n}{\partial \theta_i} = \frac{\partial}{\partial \theta_i} \left(\frac{\sin \theta_i}{\sin \theta_r}\right) and ∂n∂θr=∂∂θr(sin⁡θisin⁡θr) \frac{\partial n}{\partial \theta_r} = \frac{\partial}{\partial \theta_r} \left(\frac{\sin \theta_i}{\sin \theta_r}\right)
d
Substitute the values of the partial derivatives and the uncertainties of the variables into the uncertainty propagation formula to find Δn\Delta n
e
Report the refractive index along with its uncertainty in the correct format: n±Δnn \pm \Delta n
Answer
The refractive index of the given prism is calculated along with its uncertainty using the law of uncertainty propagation.
Key Concept
Uncertainty Propagation: The uncertainty in a calculated quantity that depends on multiple variables can be determined using the law of uncertainty propagation. This involves calculating the partial derivatives of the quantity with respect to each variable and combining them with the uncertainties of the variables.
Explanation
By applying the law of uncertainty propagation, we can determine the uncertainty in the refractive index based on the uncertainties in the measured variables.
Calculate the uncertainty of the refractive index of the given prism using the law of uncertainty propagation. Note obtained result along with the uncertainty in the correct format.
Solution
a
Identify the variables involved in the calculation of the refractive index and their uncertainties. Let’s denote the refractive index as nn, and the variables involved as x1,x2,…,xnx_1, x_2, \ldots, x_n with their respective uncertainties Δx1,Δx2,…,Δxn\Delta x_1, \Delta x_2, \ldots, \Delta x_n
b
Use the law of uncertainty propagation to calculate the uncertainty in the refractive index. The general formula for uncertainty propagation is: Δn=(∂n∂x1Δx1)2+(∂n∂x2Δx2)2+…+(∂n∂xnΔxn)2\Delta n = \sqrt{\left(\frac{\partial n}{\partial x_1} \Delta x_1\right)^2 + \left(\frac{\partial n}{\partial x_2} \Delta x_2\right)^2 + \ldots + \left(\frac{\partial n}{\partial x_n} \Delta x_n\right)^2}
c
Calculate the partial derivatives of the refractive index with respect to each variable. For example, if nn is a function of x1x_1 and x2x_2, then: ∂n∂x1 and ∂n∂x2\frac{\partial n}{\partial x_1} \text{ and } \frac{\partial n}{\partial x_2} Substitute these partial derivatives and the uncertainties into the formula from step b
d
Compute the final uncertainty Δn\Delta n and express the refractive index with its uncertainty in the correct format: n±Δnn \pm \Delta n
Answer
The uncertainty in the refractive index of the given prism is calculated along with its uncertainty using the law of uncertainty propagation.
Key Concept
Law of Uncertainty Propagation: The uncertainty in a calculated quantity that depends on multiple variables can be determined by combining the uncertainties of the individual variables. The general formula is: Δf=(∂f∂x1Δx1)2+(∂f∂x2Δx2)2+…+(∂f∂xnΔxn)2\Delta f = \sqrt{\left(\frac{\partial f}{\partial x_1} \Delta x_1\right)^2 + \left(\frac{\partial f}{\partial x_2} \Delta x_2\right)^2 + \ldots + \left(\frac{\partial f}{\partial x_n} \Delta x_n\right)^2}
Explanation
By identifying the variables and their uncertainties, calculating the partial derivatives, and applying the law of uncertainty propagation, we can determine the uncertainty in the refractive index.
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