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New question posted
4 months ago
New answer posted
4 months agoContributor-Level 10
For series combination: s = R? + R?
For parallel combination: p = (R? ) / (R? + R? )
Given the condition s = np:
R? + R? = n * (R? ) / (R? + R? )
(R? + R? )² = nR? R?
R? ² + 2R? R? + R? ² = nR? R?
R? ² - 2R? R? + R? ² + 4R? R? = nR? R?
(R? - R? )² = (n - 4)R?
(R? - R? )² / (R? ) = n - 4
n = 4 + (R? - R? )² / (R? )
Since (R? - R? )² is always non-negative, the minimum value of the term (R? - R? )² / (R? ) is 0. This occurs when R? = R?
Therefore, the minimum value of n is 4.
New answer posted
4 months ago
Contributor-Level 10
For the combined system of mass M and m, the acceleration under an applied force F is:
a = F / (M + m)
The static friction force (f_s) on the top block (m) provides its acceleration:
f_s = MA = m * [F / (M + m)] = mF / (M + m)
For the top block not to slip, the required static friction must be less than or equal to the maximum possible static friction (μmg):
f_s ≤ μmg
mF / (M + m) ≤ μmg
F ≤ μ (M + m)g
Using the values implied in the solution:
F ≤ 21 N
New answer posted
4 months agoContributor-Level 10
The force (F) exerted by radiation is the rate of change of momentum (p).
F = Δp/Δt
For photons, p = E/c. So, F = (1/c) * (ΔE/Δt).
Since Power (P) is ΔE/Δt, F = P/c.
Intensity (I) is Power per unit Area (P/A).
The formula provided in the document is F/A = (nE)/ (Δt c A) which leads to a final calculated value of 25 W/cm².
New answer posted
4 months agoContributor-Level 10
The formula for escape velocity (v_e) is v_e = √ (2GM/R).
According to the question, the new escape velocity (v_e') from a new radius R' is related to the original escape velocity by 10v_e' = v_e.
10 * √ (2GM/R') = √ (2GM/R)
Squaring both sides:
100 * (2GM/R') = (2GM/R)
100/R' = 1/R
R' = R/100
If R is the radius of Earth (6400 km), then:
R' = 6400 km / 100 = 64 km
New answer posted
4 months agoContributor-Level 10
The number of revolutions can be found using the rotational kinematic equation for angular displacement (θ):
θ = (ω_initial + ω_final)/2 * t
Number of revolutions = θ / 2π
Number of revolutions = [ (ω_final + ω_initial) * t] / (2 * 2π)
Based on the numerical values provided in the document, the calculation is:
Number of revolution = [ (2π * 3360/60 + 0) * t] / (2 * 2π) . with further calculation yielding the result:
Number of revolution = 728
New answer posted
4 months ago
Contributor-Level 10
The equations for an object rolling down an inclined plane without slipping are:
· Force equation: mg sinθ - f_s = ma
· Torque equation: f_s R = Iα
Since a = αR, we can write f_s = Iα/R = Ia/R².
Substituting this into the force equation:
mg sinθ - Ia/R² = ma
mg sinθ = a (m + I/R²)
a = (mg sinθ) / (m + I/R²)
The time taken to travel a distance S is given by S = ½ at², which means t ∝ 1/√a. Therefore, the object with the largest acceleration (a) will arrive first.
The problem is analyzed for different bodies:
· Ring: I =
New answer posted
4 months agoContributor-Level 10
Initial charge Q = CV = 14 * 10? ¹² * 12 = 168 * 10? ¹² C
Initial energy U_in = ½ CV² = ½ (14 * 10? ¹²) * 12² = 1008 pJ
When the battery is disconnected and a dielectric (k=7) is inserted, the new capacitance is C' = kC.
The charge Q remains constant.
Final energy U_f = Q²/2C' = Q²/ (2kC) = (CV)²/ (2kC) = CV²/ (2k)
U_f = (14 * 10? ¹² * 12²) / (2 * 7) = 144 pJ
Mechanical energy available for oscillation
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