Find time taken by the block to slide a distance \(8.8\,\text{m}\) with respect to the wedge (in seconds). All surfaces are smooth.
Find time taken by the block to slide a distance \(8.8\,\text{m}\) with respect to the wedge (in seconds). All surfaces are smooth.
- \(1.89\)
- \(1.29\)
- \(1.76\)
- \(2\)
The Correct Option is A
Solution and Explanation
Step 1: Equation for whole system Let acceleration of wedge be \(a\). For horizontal motion of system: \[ 24 = 10a + 2(a + a_r\cos37^\circ) \] \[ 24 = 12a + 2a_r\cos37^\circ \] Using \(\cos37^\circ = \frac{4}{5}\) \[ 24 = 12a + \frac{8a_r}{5} \] \[ 24 = 12a + \frac{8a_r}{5} \quad ...(1) \]
Step 2: Acceleration of block relative to wedge Along the incline: \[ mg\sin37^\circ - ma\cos37^\circ = ma_r \] \[ 2(10)\frac{3}{5} - 2a\frac{4}{5} = 2a_r \] \[ 6 - \frac{4a}{5} = a_r \quad ...(2) \] Step 3: Solve equations Substitute (2) into (1): \[ 24 = 12a + \frac{8}{5}\left(6 - \frac{4a}{5}\right) \] Solving, \[ a_r = 4.92\ \text{m/s}^2 \] Step 4: Time taken Relative displacement: \[ s = \frac12 a_r t^2 \] \[ 8.8 = \frac12 (4.92)t^2 \] \[ t^2 = 3.58 \] \[ t = 1.89\ \text{s} \]
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