Force and Translational Dynamics

A force is an interaction exerted by another object. Mass measures inertia; weight is gravitational force. A free-body diagram includes forces acting on the chosen system, not every force in the surrounding situation.

When object A exerts a force on B, B exerts an equal-magnitude opposite-direction force on A. The pair acts on different objects, so the two forces do not cancel on either individual free-body diagram. A truck and a small car exert equal contact forces on each other during a collision even though their accelerations can differ.

A normal force is perpendicular to the contact surface, not automatically vertically upward or equal to mg. On an incline, resolve weight into components parallel and perpendicular to the surface. A rope’s idealized tension may be uniform if the rope is massless and pulleys are ideal; otherwise additional modeling is needed.

Static friction can take values from zero up to a maximum; it is not always equal to μsN. It acts to oppose the relative slipping tendency at the contact. Kinetic friction in the simple model has magnitude μkN and opposes relative sliding. Determine whether slipping occurs before choosing the friction relationship.

Friction can accelerate an object in its direction of motion, as when a conveyor belt speeds up a box. The statement “friction always points backward” is therefore unreliable. Use relative contact motion, not just the object’s velocity relative to the room.

An object moving at constant speed in a circle still accelerates because its velocity direction changes. The radial acceleration has magnitude v²r and points toward the center. “Centripetal force” names the inward net-force requirement; it is not an additional physical interaction to add beside tension or gravity.

Separate radial and tangential directions when speed changes. The radial net force controls curvature, while a tangential component changes speed. In a vertical circle, gravity’s radial contribution changes with position, so tension or normal force need not be constant.

Newton’s gravitational model gives an attractive force proportional to the product of masses and inversely proportional to center-to-center separation squared. Near Earth’s surface, g can be treated as nearly constant over modest height changes. At orbital distances, use the distance from Earth’s center rather than height above the ground alone.

An orbiting object is in free fall and still experiences gravity. Apparent weightlessness can result from the absence of a supporting normal force, not the absence of gravitational force. For a circular orbit, gravity can supply the required inward acceleration.

For motion on an incline, axes parallel and perpendicular to the surface often simplify the equations. Weight has components mg sinθ parallel to the slope and mg cosθ perpendicular to it when θ is the incline angle. The normal force is perpendicular to the surface and need not equal mg. It equals the appropriate value required by the perpendicular acceleration equation.

Write a separate net-force equation for each axis. If the object has no acceleration perpendicular to the incline, that component’s net force is zero. Friction acts along the surface and opposes relative sliding or its tendency; it does not always oppose the object’s velocity relative to the ground. Check which surfaces interact before assigning its direction.

A massless inextensible string over an ideal pulley can impose a common acceleration magnitude on connected objects. Analyze each object with its own force diagram or choose the whole system to eliminate internal tension from the net-force equation. The tension can then be recovered from an individual object’s equation.

Equal tension on both sides is an idealization that may fail for a pulley with significant rotational inertia. Newton’s third-law pairs act on different objects and have equal magnitude even when the objects have different accelerations. A large and small object interacting do not exert unequal forces merely because one moves more dramatically. Their acceleration magnitudes differ because their masses differ.

The inward acceleration required for circular motion is v²r. The net inward force can be supplied by tension, gravity, friction, a normal force, or components of several forces. Do not add a separate “centripetal force” after already drawing the physical interactions; centripetal describes the direction of the net force.

For an orbit, gravity can provide the needed inward acceleration. For a car turning on a level road, static friction can supply it. At the top of a vertical circle, inward points downward; at the bottom, inward points upward. Choose the radial direction at the actual location before assigning signs, and distinguish changes in direction from changes in speed.

Identify the object or system and draw only interactions exerted on it. Weight is the gravitational interaction, the normal force is perpendicular to a contact surface, tension acts along an ideal taut cord, and friction is tangent to contact. Velocity and acceleration are not extra forces. The force the object exerts on another object belongs on the other object’s diagram.

Newton’s third-law pair acts on different objects and therefore does not cancel on one object’s diagram. For a book on a table, the book’s weight and the table’s normal force can balance, but they are not a third-law pair. The gravitational partner is the book’s force on Earth.

Write component equations after choosing axes. A normal force is not automatically mg: other vertical forces or vertical acceleration can change it. Static friction adjusts up to a maximum; it is not always equal to μsN. Determine the required value from the motion or equilibrium conditions, then compare it with the limit.

For constant mass, net force determines acceleration through ΣF=ma. Zero net force permits rest or constant velocity; it does not require zero velocity. For connected objects, draw separate diagrams when internal forces such as tension are requested. A combined-system equation can eliminate internal forces and simplify the acceleration calculation.

Ideal massless cords and frictionless massless pulleys impose useful constraints, but do not silently apply those constraints to a massive pulley. On an incline, resolve gravity parallel and perpendicular to the surface. The component down the slope is mg sin θ when θ is measured from horizontal; the perpendicular component is mg cos θ.

A force model should reproduce observed direction and magnitude. If a computed static-friction requirement exceeds its maximum, the assumed no-slip state is not feasible. If an acceleration sign is negative, interpret it relative to your chosen positive direction rather than discarding the result.

An object moving around a circle changes velocity direction even at constant speed. The radial acceleration magnitude is v²r. “Centripetal force” names the net inward component of the actual forces; it is not an additional interaction to add beside tension, gravity or a normal force.

At the top and bottom of a vertical circle, write inward as the radial direction for each location and resolve the actual forces accordingly. A contact force can vanish at the threshold of losing contact. Circular-motion equations do not guarantee that the object can remain constrained to the circle under every speed.

For gravitational circular motion, equate the gravitational force to the required radial net force under the simplified two-body assumptions. Mass can cancel for the orbiting object. Check that radius is measured from the central body’s center rather than from its surface.

Choose one object or a clearly defined system and draw only external forces acting on it. Weight is Earth’s force on the object; a normal force is a contact force perpendicular to a surface; friction is tangential contact force. Do not add a separate “force of motion” or treat ma as another physical force.

Newton’s third-law partners act on different objects, so they do not cancel in one object’s force sum. Within a larger combined system, internal force pairs cancel in the net external force calculation. This is a reason to choose a combined system when internal tensions are not the target.

Resolve forces along useful axes, often parallel and perpendicular to an incline. The normal force is not automatically mg; it follows from the perpendicular force equation and acceleration. A sloped surface, an extra applied force, or vertical acceleration changes it.

Centripetal acceleration points toward the center and has magnitude v²r. The net inward component of actual forces must equal mv²r. “Centripetal force” names this resultant requirement, not a new force to add alongside tension, gravity, or normal force.

In a vertical circle, gravity’s radial contribution changes with position. At the top, it can point toward the center; at the bottom, away from it. Write the radial equation separately at each location using a consistent inward-positive direction. The minimum speed for maintaining contact depends on the physical constraint.

Uniform circular motion has constant speed but changing velocity direction. Tangential acceleration changes speed; radial acceleration changes direction. A force can have both components. A curved path alone does not tell you that the force magnitude is constant.