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$$N + P \sin \theta = mg$$

Datos: $P = 50~\mathrm{N}$, $\theta = 30^\circ$, $mg = 100~\mathrm{N}$.

$$\sin 30^\circ = \frac{1}{2}$$ $$P \sin \theta = 50 \times \frac{1}{2} = 25~\mathrm{N}$$ $$N = 100 - 25 = 75~\mathrm{N}$$

$N = 75~\mathrm{N}$

Parte (c): Calcular la aceleración $a$

$$\cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866$$ $$P \cos \theta = 50 \times 0.866 = 43.3~\mathrm{N}$$

$$P \cos \theta - f_k = ma$$ $$43.3 - 22.5 = 10a$$ $$20.8 = 10a$$ $$a = 2.08~\mathrm{m/s^2}$$

$a = 2.08~\mathrm{m/s^2}$

Parte 2: Si $P$ se duplica ($P=100~\mathrm{N}$), $\mu_s=0.5$, $\mu_k=0.3$, calcular $a$

$$N + 100 \sin 30^\circ = 100$$ $$100 \times \frac{1}{2} = 50~\mathrm{N}$$ $$N = 100 - 50 = 50~\mathrm{N}$$

$$f_{s,\max} = \mu_s N = 0.5 \times 50 = 25~\mathrm{N}$$

Componente horizontal: $100 \cos 30^\circ = 100 \times 0.866 = 86.6~\mathrm{N} > 25~\mathrm{N}$

Sí se mueve (usa fricción cinética).

$$86.6 - 15 = 10a$$ $$71.6 = 10a$$ $$a = 7.16~\mathrm{m/s^2}$$

$a = 7.16~\mathrm{m/s^2}$