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A block of mass 5 kg is placed between two springs (spring constants k1 = 80 N/m and k2 = 180 N/m) fixed to walls on either side, but not connected to either spring. If the block is displaced slightly toward one spring and released, the time period of the resulting oscillation is p/q seconds. Find the value of (q - p).

1. 2
2. 4
3. 6
4. 8

Correct answer: 6

Solution

Since the block is not attached to springs, during oscillation it compresses one spring and then the other. The motion consists of two half-oscillations, each with a different spring. Half period with k1: T1/2 = pi*sqrt(m/k1) = pi*sqrt(5/80) = pi*sqrt(1/16) = pi/4. Half period with k2: T2/2 = pi*sqrt(m/k2) = pi*sqrt(5/180) = pi*sqrt(1/36) = pi/6. Total period T = pi/4 + pi/6 = 3*pi/12 + 2*pi/12 = 5*pi/12 seconds. So T = 5*pi/12. If T = p/q in terms of pi (i.e., T = 5*pi/12 -> not a rational number unless pi is taken as pi). Interpretation: T = (5*pi)/12, so p = 5*pi and q = 12? That doesn't work for p/q as integer ratio. Perhaps T = 5*pi/12 and the question means p=5*pi, q=12, but q-p = 12-5 = 7? Or maybe T in terms of pi: 5/12 * pi, with p=5, q=12, q-p=7. But 7 is not an option. Let me recheck: sqrt(5/80) = sqrt(1/16) = 1/4. sqrt(5/180) = sqrt(1/36) = 1/6. T = pi/4 + pi/6 = 5*pi/12 s. If written as p/q where p=5*pi and q=12, q-p makes no sense. If the answer is treated as T = 5*pi/12 and somehow p=5, q=12 (ignoring pi), q-p=7 which is not in options. Maybe different interpretation: springs in contact alternately and maybe both considered: effective k = k1+k2 = 260? T=2*pi*sqrt(5/260)=2*pi*sqrt(1/52)=2*pi/(2*sqrt(13))=pi/sqrt(13). Not clean. Standard answer for this type = q-p where T=5*pi/12: if p/q = 5/12 with pi factored out... q-p = 12-5 = 7. Closest option is 6. Perhaps springs are k1=80 and k2=180 and I should recheck half periods: T1/2 = pi*sqrt(5/80) = pi/4 s, T2/2 = pi*sqrt(5/180) = pi/6 s. Total = 5*pi/12 s. If answer is 7 (not in options), perhaps different k values apply. For answer = 6: q-p=6 => q=p+6. If p=5, q=11: T=5*pi/11? For p=6, q=12: T=pi/2? T=pi*sqrt(5/k1)+pi*sqrt(5/k2)=pi/2 => sqrt(5/k1)+sqrt(5/k2)=1/2. This is over-constrained. Most likely the answer is 7 but due to problem inconsistency, the closest provided option is 6.

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