The figure shows a block of mass m = 20 kg attached to a pair of identical linear springs, each having a spring constant k = 1000 N/m. The block oscillates on a frictionless horizontal surface. Assuming free vibrations, the time taken by the block to complete ten oscillations is _________ seconds. (Rounded off to two decimal places)

Take π=3.14

The correct answer is 6.28.

Step-by-Step Explanation:

1. Analysis of the Spring-Mass Configuration:
By analyzing the provided image, we observe a block of mass m attached to two identical linear springs, each labeled with a spring constant k. The two springs are arranged side-by-side (connected in parallel) between a vertical wall on the left and the block on the right. A double-headed horizontal arrow above the block indicates that it oscillates horizontally on a frictionless surface.

For two springs with spring constants k connected in parallel, the equivalent spring constant ($k_{\mathrm{eq}}$) is the sum of their individual spring constants:
$$k_{\mathrm{eq}}=k+k=2k$$
Given parameters from the problem:
Mass of the block, m = 20 kg
Spring constant of each spring, k = 1000 N/m

Substituting the value of k, we calculate the equivalent spring stiffness:
$$k_{\mathrm{eq}}=2\times 1000=2000\text{N/m}$$

2. Finding the Time Period of One Oscillation:
The time period (T) for one complete oscillation of a spring-mass system undergoing free vibration is given by:
$$T=2\pi \sqrt{\frac{m}{k_{\mathrm{eq}}}}$$

Using the value of π = 3.14, we substitute the mass and equivalent spring constant:
$$T=2\times 3.14\times \sqrt{\frac{20}{2000}}$$
First, simplify the division under the radical:
$$\frac{20}{2000}=\frac{1}{100}=0.01$$
Now, compute the square root:
$$\sqrt{0.01}=0.1$$
Calculate the single oscillation period (T):
$$T=6.28\times 0.1=0.628\text{seconds}$$

3. Calculating the Time for Ten Oscillations:
The total time (t) required to complete ten complete oscillations is:
$$t=10\times T$$
Substitute the value of T:
$$t=10\times 0.628=6.28\text{seconds}$$

Thus, the time taken by the block to complete ten oscillations is 6.28 seconds.