A recipe that never fails
The inclined plane
A block on a ramp is the classic test. The trick is the axis choice: instead of horizontal and vertical, tilt your axes so one runs along the inclined plane and the other perpendicular to it. Now the acceleration lies entirely along one axis, and only gravity needs resolving.
Perpendicular axis: the normal force balances mg·cos(theta). Along the slope with no friction: acceleration is g·sin(theta), independent of mass.
Add kinetic friction and it acts up the slope, opposing the slide, with magnitude \mu_k N = \mu_k mg\cos\theta. The along-slope equation becomes mg\sin\theta - \mu_k mg\cos\theta = ma, so a = g(\sin\theta - \mu_k\cos\theta). For \theta = 30^\circ and \mu_k = 0.20: a = 9.8\,(0.500 - 0.20\times 0.866) = 9.8\times 0.327 \approx 3.2\ \text{m/s}^2 down the ramp.
Connected systems: the Atwood machine
Now two objects share one string over a pulley — the Atwood machine. They are forced to move together, so they share a single acceleration magnitude and (for an ideal string and pulley) a single tension. Draw a free-body diagram for each mass, take "toward the heavier side" as positive, and write the second law for both. Adding the equations eliminates the tension.
Acceleration of the Atwood system and the tension in the string, with m_2 the heavier mass.
The results pass every sanity check. If the masses are equal, a = 0 and T = mg — the system just hangs balanced. If one mass is zero, a = g — the other is in free fall. And the tension always sits between the two weights, which is why an Atwood machine is a gentle, easily-timed way to measure g.
Circular motion: centripetal force is not a new force
Whirl a ball on a string in a circle at steady speed. Its speed is constant, yet its velocity — a vector — is constantly turning, so it is accelerating, pointed toward the centre. By the second law there must be a net force toward the centre too, called the centripetal force. For uniform circular motion its size is fixed by the speed and radius.
The net inward (centripetal) force needed to keep a mass moving in a circle of radius r at speed v.
Where Newton's laws lead next
You can now analyse forces on almost any everyday object. Two powerful shortcuts grow directly out of these laws. Multiply force by the distance over which it acts and you reach work and energy, and the work-energy theorem — often solving in one line what forces solve in five. Multiply force by the time over which it acts and you reach impulse and momentum, whose conservation cracks open collisions and rockets.
The same three laws, rewritten for spinning bodies, become torque, moment of inertia and angular momentum. Newton handed us a single machine — isolate the object, add the forces, set the sum equal to ma — and the rest of classical mechanics is that machine, turned to ever richer problems.