A block of mass $5\,\text{kg}$ moves along the $x$-direction subject to the force $F = (-20x + 10)\,\text{N}$, with the value of $x$ in metre. At time $t = 0\,\text{s}$, it is at rest at position $x = 1\,\text{m}$. The position and momentum of the block at $t = \frac{\pi}{4}\,\text{s}$ are

Step 1: Identify the given parameters.

The mass of the block and the force acting on it are given.

$$m = 5\,\text{kg}$$ $$F = -20x + 10$$

At $t = 0\,\text{s}$, the block is at rest.

$$x(0) = 1\,\text{m}$$ $$v(0) = 0\,\text{m/s}$$

Step 2: Determine the equilibrium position.

The equilibrium position $x_0$ occurs where the net force is zero.

$$-20x_0 + 10 = 0$$ $$x_0 = 0.5\,\text{m}$$

Step 3: Identify the parameters of the Simple Harmonic Motion (SHM).

Express the force in terms of the displacement from equilibrium.

$$F = -20(x - 0.5)$$

Comparing this to the standard SHM force equation $F = -k(x - x_0)$ yields the spring constant.

$$k = 20\,\text{N/m}$$

Calculate the angular frequency $\omega$.

$$\omega = \sqrt{\frac{k}{m}}$$ $$\omega = \sqrt{\frac{20}{5}} = 2\,\text{rad/s}$$

Step 4: Determine the amplitude and phase.

The general equation for the position in SHM is given by:

$$x(t) = x_0 + A\cos(\omega t + \phi)$$

Since the block starts from rest at $t = 0$, the phase angle is zero.

$$\phi = 0$$

Using the initial position at $t = 0$:

$$1 = 0.5 + A\cos(0)$$ $$A = 0.5\,\text{m}$$

Step 5: Calculate the position at $t = \frac{\pi}{4}\,\text{s}$.

Substitute the parameters into the position equation.

$$x(t) = 0.5 + 0.5\cos(2t)$$

Evaluate at $t = \frac{\pi}{4}\,\text{s}$.

$$x\left(\frac{\pi}{4}\right) = 0.5 + 0.5\cos\left(\frac{\pi}{2}\right)$$ $$x\left(\frac{\pi}{4}\right) = 0.5\,\text{m}$$

Step 6: Calculate the momentum at $t = \frac{\pi}{4}\,\text{s}$.

Differentiate the position equation to find the velocity.

$$v(t) = -A\omega\sin(\omega t)$$ $$v(t) = -0.5(2)\sin(2t)$$ $$v(t) = -\sin(2t)$$

Evaluate the velocity at $t = \frac{\pi}{4}\,\text{s}$.

$$v\left(\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{2}\right) = -1\,\text{m/s}$$

Calculate the momentum $p = mv$.

$$p = 5 \cdot (-1)$$ $$p = -5\,\text{kg}\cdot\text{m/s}$$

Answer: Position is $0.5\,\text{m}$ and momentum is $-5\,\text{kg}\cdot\text{m/s}$.