Vertical Circular Motion

When an object moves in a vertical circle, the forces acting on it vary with position. Unlike horizontal circular motion where the speed remains constant, the speed in vertical circular motion continuously changes because gravity affects the object's motion as it moves up and down.

Watch how the velocity vector (green arrow) in the animation below changes length as the cart moves around the loop—it's longest at the bottom and shortest at the top. Gravity $mg$ (brown arrow) always points downward, while the normal force $N$ (teal arrow) changes both direction and magnitude as the cart moves.

At the bottom of the loop:

The cart moves fastest at the bottom. Normal force $N$ points upward (toward center), gravity $mg$ points downward (away from center). Net force toward center:

$$\sum F_c = N - mg = \frac{mv^2}{r}$$

Solving for $N$:

$$N = mg + \frac{mv^2}{r}$$

At the top of the loop:

The cart moves slowest at the top. Both $N$ and $mg$ point downward (toward center). Net force toward center:

$$\sum F_c = N + mg = \frac{mv^2}{r}$$

Solving for $N$:

$$N = \frac{mv^2}{r} - mg$$

Gravity assists in providing centripetal force, so $N$ is smaller than at the bottom.

At the side of the loop:

Normal force $N$ points horizontally toward center. Gravity $mg$ points downward (perpendicular to the radial direction). Only $N$ contributes to centripetal force:

$$\sum F_c = N = \frac{mv^2}{r}$$

However, as the cart descends from the top toward the left side (for counterclockwise motion), gravity has a component along the direction of motion, causing the cart to speed up.

We can understand this by looking at the free body diagrams below and seeing that $\vec{F}_{net}$ doesn't point solely in the centripetal direction. It also has a tangential component (due to gravity), which, by Newton's Second Law, means there must be acceleration in that direction.

On the opposite side (right side of the loop), the forces are mirrored: $N$ points toward the center (to the left), and $mg$ still points downward.

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