A particle executes SHM such that at a given time it is at x= +1/4 the amplitude, moving away from equilibrium, and 0.7 seconds later the particle has -1/6 the maximum speed moving away from equilibrium. Find the period of the motion.

To find the period of the motion, we need to gather some information about the Simple Harmonic Motion (SHM) of the particle.

Let's consider the given information:
At a certain time, the particle is at x = 1/4 of the amplitude, moving away from equilibrium. This indicates that the particle is at x = (1/4)A, where A represents the amplitude of the motion.

0.7 seconds later, the particle has a velocity of -1/6 times the maximum speed, also moving away from equilibrium. This implies that the particle has a velocity v = -(1/6)v_max, where v_max represents the maximum velocity of the motion.

In SHM, the displacement x of the particle as a function of time t can be represented by:
x = A * cos(2πt / T),

where T represents the period of the motion.

The velocity v of the particle as a function of time t can be represented by:
v = -A * (2π / T) * sin(2πt / T),

where the negative sign indicates that the particle is moving in the opposite direction to the displacement.

Now, let's use this information to find the period of the motion.

Given that the particle is at x = (1/4)A and v = -(1/6)v_max at two different times, we can set up two equations based on the above formulas:

When t = 0: x = (1/4)A = A * cos(0) = A,
and v = 0.

When t = 0.7 seconds: x = -A * (1/4) = A * cos(2π(0.7) / T),
and v = -(1/6)v_max = -A * (2π / T) * sin(2π(0.7) / T).

Now, to solve for the period T, we'll use the equation for x at t = 0.7 seconds:

-(1/4) = cos(2π(0.7) / T).

To determine the period T, we need to find the value of T that satisfies this equation. However, it's important to note that this equation cannot be solved algebraically. We will need to use numerical methods or approximations to find the value of T.

You can use numerical methods like Newton's method or try approximating it using a graphing calculator or software.