Problem Statement
Solve the work-energy problem: A diffraction grating with $N = 1200$ slits is used in 2nd order. Find the resolving power and the minimum resolvable wavelength difference near $\lambda = 500$ nm. $$\mathcal{R} = mN = 2 \times 1200 = \boxed{2400}$$ $$\delta\lambda_{min} = \frac{\lambda}{\mathcal{R}} = \frac{500}{2400} \approx \box
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$\mathcal{R} = mN = 2 \times 1200 = \boxed{2400}$$
$$\delta\lambda_{min} = \frac{\lambda}{\mathcal{R}} = \frac{500}{2400} \approx \box
Given Information
- Mass $m$, velocity $v$, height $h$, or other given quantities
- Any forces doing work (conservative or non-conservative) as specified
Physical Concepts & Formulas
The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy: $W_{\text{net}} = \Delta KE$. For conservative forces (gravity, spring, electric), a potential energy function $U$ exists such that $W = -\Delta U$, and the total mechanical energy $E = KE + U$ is conserved. Non-conservative forces (friction, air drag) remove mechanical energy, converting it to thermal energy. The power delivered is $P = dW/dt = \vec{F}\cdot\vec{v}$.
- $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$ — work done by constant force
- $KE = \frac{1}{2}mv^2$ — kinetic energy
- $U_g = mgh$ — gravitational PE (near Earth’s surface)
- $U_s = \frac{1}{2}kx^2$ — elastic PE
- $W_{\text{net}} = \Delta KE = KE_f – KE_i$ — work-energy theorem
- $E_i = E_f$ (when only conservative forces act)
Step-by-Step Solution
Step 1 — Identify all forces and whether they are conservative.
Step 2 — Apply conservation of energy (if no friction):
$$
$$
Step 3 — If friction acts:
$$
Answer
$$\mathcal{R} = mN = 2 \times 1200 = \boxed{2400}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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