Velocity, Acceleration, and Net Force in SHM 

When a spring oscillates back and forth, or a pendulum swings through its arc, the motion appears smooth and rhythmic. However, beneath that motion, three quantities are constantly changing: velocity, acceleration, and net force. Understanding how these three interact is essential to understanding Simple Harmonic Motion.

In SHM, the defining rule is that the net force always points toward equilibrium and is proportional to displacement. Because acceleration depends directly on net force through Newton’s Second Law, acceleration also always points toward equilibrium. Velocity behaves differently. Velocity depends on the direction the object is moving, not just its position. This distinction is what makes SHM conceptually interesting.

Concept Explanation

As an object moves away from equilibrium, the restoring force increases in magnitude. This means acceleration increases as displacement increases. At maximum displacement, the force and acceleration are greatest. However, at that same point, velocity is zero because the object is changing direction. As the object moves back toward equilibrium, the restoring force decreases, which means the acceleration also decreases. Meanwhile, velocity increases because the object has been accelerating toward the center. When the object reaches equilibrium, velocity is at its maximum, while net force and acceleration are momentarily zero. After passing equilibrium, the force reverses direction and begins slowing the object down until it reaches maximum displacement on the opposite side. This repeating pattern defines SHM.

Motion in a Mass–Spring System

Consider a mass attached to a horizontal spring on a frictionless surface. The equilibrium position is where the spring is neither stretched nor compressed. When the mass is pulled to the right and released, it begins oscillating. At the maximum rightward displacement, the mass momentarily stops before reversing direction. Its velocity is zero at that instant. However, the spring is stretched the most at this position, so the restoring force is at its largest magnitude. The net force acts to the left, pulling the system back toward equilibrium, and the acceleration also points to the left.

As the mass moves toward equilibrium, the displacement decreases, so the restoring force decreases as well. However, the mass speeds up because it has been accelerating toward the center. By the time the mass passes through equilibrium, the displacement is zero. This means the net force is zero and the acceleration is zero at that instant. Yet the velocity is maximum because acceleration has been building its speed.

After passing the equilibrium, the spring begins compressing. The restoring force switches direction and now acts opposite the motion, slowing the mass down. When the mass reaches maximum left displacement, velocity is zero again, while force and acceleration are maximum toward equilibrium. The motion is symmetric and continues repeating. This would look like the first diagram, but the net force and acceleration point in the opposite direction.

Motion in a Pendulum

A pendulum follows the same overall pattern when the angle is small. The equilibrium position is the lowest point of the swing. Gravity always acts downward, but only the tangential component of gravity acts as the restoring force that pulls the bob toward equilibrium.

At the maximum angle, the pendulum bob momentarily stops before reversing direction. Its velocity is zero at that instant. However, the tangential component of gravity is greatest there, so the restoring force and acceleration are also greatest and directed toward the lowest point.

As the pendulum swings toward the lowest point, the restoring force decreases while velocity increases. At the bottom of the swing, the tangential component of gravity becomes zero. Therefore, acceleration and net force are zero at that instant, while velocity is maximum.

Real World Example 

When an elevator begins to move upward, you feel a brief sensation of extra weight. At that moment, your velocity is still zero because the elevator has not yet picked up speed, but the floor is pushing upward harder than usual. This creates a net upward force, which means your acceleration is also upward and at its maximum. Even though you are not yet moving, the force and acceleration are strongest right at the start of the motion.

As the elevator continues upward, the force from the floor decreases, and the acceleration gradually drops to zero. By the time the elevator reaches its cruising speed, the net force has become zero and the acceleration has vanished entirely. Yet this is the moment when your velocity is at its maximum. The elevator is moving steadily, and because there is no net force acting on you, you feel completely normal again.

When the elevator approaches the top floor and begins to slow down, the situation reverses. The floor now pushes up less than your weight, creating a net downward force. Your acceleration is downward, even though you are still moving upward. This is why you briefly feel lighter. As the elevator comes to a stop, your velocity drops to zero once again, while the net force and acceleration reach their maximum (downward) values for a moment before returning to normal.

Why This Matters

In springs, pendulums, and in real systems, the pattern remains consistent. At maximum displacement, velocity is zero while force and acceleration are greatest toward equilibrium. At equilibrium, velocity is maximum while force and acceleration are zero. Acceleration depends on displacement, not on velocity. Recognizing this pattern allows us to analyze any point in the motion and determine exactly what the system is doing.