A simple pendulum 4 m long swings with an amplitude of 0.2m. Compute the velocity of the pendulum at its lower point and its acceleration at extreme end.

$$
\omega = \sqrt{\frac{g}{L}}
$$
$$
\omega = \sqrt{\frac{9.8 \text{ m/s}^2}{4 \text{ m}}}
$$
$$
\omega = \sqrt{2.45 \text{ s}^{-2}}
$$
$$
\omega \approx 1.565 \text{ rad/s}
$$

$$
v_{max} = A\omega
$$
$$
v_{max} = (0.2 \text{ m})(1.565 \text{ rad/s})
$$
$$
v_{max} \approx 0.313 \text{ m/s}
$$
The velocity of the pendulum at its lower point is $\boxed{\text{0.313 m/s}}$.

$$
a_{max} = A\omega^2
$$
$$
a_{max} = (0.2 \text{ m})(1.565 \text{ rad/s})^2
$$
$$
a_{max} = (0.2 \text{ m})(2.45 \text{ s}^{-2})
$$
$$
a_{max} = 0.49 \text{ m/s}^2
$$
The acceleration of the pendulum at the extreme end is $\boxed{\text{0.49 m/s}^2}$.

$$
B_H = B \cos \delta
$$

$$
B_H = (4.5 \times 10^{-7} \text{ T}) \cos(30^\circ)
$$
$$
B_H = (4.5 \times 10^{-7} \text{ T}) \left(\frac{\sqrt{3}}{2}\right)
$$
$$
B_H = (4.5 \times 10^{-7} \text{ T})(0.866)
$$
$$
B_H \approx 3.897 \times 10^{-7} \text{ T}
$$
The horizontal component is $\boxed{3.90 \times 10^{-7} \text{ T}}$.

$$
B_V = B \sin \delta
$$

$$
B_V = (4.5 \times 10^{-7} \text{ T}) \sin(30^\circ)
$$
$$
B_V = (4.5 \times 10^{-7} \text{ T})(0.5)
$$
$$
B_V = 2.25 \times 10^{-7} \text{ T}
$$
The vertical component is $\boxed{2.25 \times 10^{-7} \text{ T}}$.

$$
\frac{1}{f} = \frac{1}{v} - \frac{1}{u}
$$

$$
\frac{1}{v} = \frac{1}{f} + \frac{1}{u}
$$

$$
\frac{1}{v} = \frac{1}{10 \text{ cm}} + \frac{1}{-15 \text{ cm}}
$$
$$
\frac{1}{v} = \frac{1}{10 \text{ cm}} - \frac{1}{15 \text{ cm}}
$$

$$
\frac{1}{v} = \frac{3}{30 \text{ cm}} - \frac{2}{30 \text{ cm}}
$$
$$
\frac{1}{v} = \frac{1}{30 \text{ cm}}
$$
$$
v = 30 \text{ cm}
$$
The image distance is $\boxed{\text{30 cm}}$. (Positive sign indicates a real image formed on the opposite side of the lens).

$$
M = \frac{v}{u}
$$

$$
M = \frac{30 \text{ cm}}{-15 \text{ cm}}
$$
$$
M = -2
$$
The magnification is $\boxed{-2}$. (Negative sign indicates an inverted image, and $|M|=2$ means it is twice the size of the object).