In a block spring system, m=0.2kg and k=5N/m. At t=pi/10 s, the spring has compression of 6cm and the block has velocity V=-40m/s. What is the first time (greater than 0) at which... In a block spring system, m=0.2kg and k=5N/m. At t=pi/10 s, the spring has compression of 6cm and the block has velocity V=-40m/s. What is the first time (greater than 0) at which the velocity is positive and 60% of it is the maximum value? Read more

Steps to Solve

1. Identify the maximum velocity In a mass-spring system, the maximum velocity ($v_{max}$) can be calculated using the formula: $$ v_{max} = A \omega $$ where $A$ is the amplitude and $\omega$ is the angular frequency.

2. Determine the desired velocity We need to find the time at which the block's velocity is 60% of the maximum velocity. Therefore, we calculate: $$ v_{desired} = 0.6 v_{max} $$

3. Write the velocity equation The velocity of a mass-spring system as a function of time can be described as: $$ v(t) = -A \omega \sin(\omega t) $$ We will set this equal to $v_{desired}$.

4. Set up the equation to solve for time Substituting the desired velocity into the velocity equation gives: $$ -A \omega \sin(\omega t) = 0.6 A \omega $$ We can simplify this by dividing both sides by $A \omega$ (assuming they are not zero): $$ -\sin(\omega t) = 0.6 $$

5. Solve for $\omega t$ Now, rearranging the equation gives: $$ \sin(\omega t) = -0.6 $$ We need to calculate the angle whose sine is -0.6. This can be found using the inverse sine function: $$ \omega t = \arcsin(-0.6) $$

6. Find the general solution Since the sine function is periodic, the general solutions for this equation can be given by: $$ \omega t = -\frac{\pi}{6} + 2\pi n \text{ or } \omega t = \frac{7\pi}{6} + 2\pi n \quad (n \in \mathbb{Z}) $$

7. Solve for time To find the first positive time, we divide by $\omega$: $$ t = \frac{-\frac{\pi}{6}}{\omega} \quad \text{or} \quad t = \frac{\frac{7\pi}{6}}{\omega} $$ The first solution will likely be negative, so we choose the second solution for the first positive time: $$ t = \frac{7\pi}{6\omega} $$