Vertical circular motion problems have a reputation for tripping up even strong physics students. Unlike horizontal circular motion where everything stays constant, vertical loops throw changing speeds, varying tensions, and gravity into the mix. But here's the good news: once you understand the pattern, these problems become remarkably predictable.
The Key Principle: Forces Change With Position
In vertical circular motion, you're dealing with an object moving in a vertical circle—think of a ball on a string being swung overhead, or a roller coaster completing a loop. The crucial insight is that the centripetal force required changes because gravity helps at some points and hinders at others.
At any point in the circle, the net force toward the centre must equal the centripetal force: F_c = mv²/r. But the forces creating this net force change depending on where you are:
- At the bottom: Tension must overcome gravity AND provide centripetal force
- At the top: Both tension and gravity point toward the centre, working together
- On the sides: Gravity acts perpendicular to the radius, tension provides all centripetal force
This is why tension is always greatest at the bottom and smallest at the top—and why the string is most likely to break at the bottom or go slack at the top.
Setting Up Your Force Equations
The systematic approach starts with clear force equations at critical points. Always draw a diagram showing the direction toward the centre of the circle.
At the bottom of the circle:
T_bottom - mg = mv²_bottom/r
T_bottom = mg + mv²_bottom/r
At the top of the circle:
T_top + mg = mv²_top/r
T_top = mv²_top/r - mg
Notice how tension and weight are on opposite sides of the equation at the bottom (tension opposes weight) but the same side at the top (both point downward toward centre).
The Critical Point: Minimum Speed at the Top
Here's where things get interesting. What happens if the object is moving too slowly at the top? The tension in the string decreases as speed decreases. At some critical minimum speed, tension drops to zero—the string goes slack. Below this speed, the object can't maintain circular motion.
At the minimum speed, set T_top = 0:
0 + mg = mv²_min/r
v_min = √(gr)
This is the most important equation for vertical circular motion problems. Below this speed at the top, circular motion is impossible.
Worked Example: Ball on a String
Let's work through a complete problem. A 0.5 kg ball is attached to a 1.2 m string and swung in a vertical circle. The ball passes through the lowest point at 8 m/s.
Part A: Find the tension at the bottom
At the bottom: T_bottom = mg + mv²/r
T_bottom = (0.5 kg)(9.8 m/s²) + (0.5 kg)(8 m/s)²/(1.2 m)
T_bottom = 4.9 N + 26.7 N = 31.6 N
Part B: Find the speed at the top
Use energy conservation. Taking the bottom as the reference level:
E_bottom = E_top
½mv²_bottom = ½mv²_top + mg(2r)
½(8)² = ½v²_top + (9.8)(2.4)
32 = ½v²_top + 23.52
v²_top = 16.96
v_top = 4.12 m/s
Part C: Find the tension at the top
T_top = mv²_top/r - mg
T_top = (0.5)(16.96)/(1.2) - (0.5)(9.8)
T_top = 7.07 - 4.9 = 2.17 N
Part D: What's the minimum speed needed at the bottom to complete the loop?
First, find minimum speed at top: v_min,top = √(gr) = √(9.8 × 1.2) = 3.43 m/s
Now use energy conservation to find the corresponding bottom speed:
½mv²_min,bottom = ½mv²_min,top + mg(2r)
½v²_min,bottom = ½(3.43)² + (9.8)(2.4)
v_min,bottom = √(11.76 + 47.04) = 7.67 m/s
Since our ball was travelling at 8 m/s at the bottom, it successfully completes the loop with some tension remaining at the top.
Common Mistakes to Avoid
Sign errors: The biggest pitfall is getting the force directions wrong. Always identify which direction is toward the centre, then determine whether each force points toward or away from centre. At the bottom, tension points up (toward centre) and weight points down (away from centre). At the top, both point down (toward centre).
Forgetting the height difference: When using energy conservation between top and bottom, the height difference is the diameter (2r), not the radius. This factor of 2 matters.
Using the wrong minimum speed: The minimum speed formula v = √(gr) applies at the top of the loop, not the bottom. If asked for minimum speed at the bottom, you must use energy conservation to work backwards from the top.
Assuming tension is constant: It's not! Tension varies continuously around the loop. You can only calculate it at specific points using force equations.
Practice Makes Permanent
The beauty of vertical circular motion problems is that once you've mastered the systematic approach—identifying critical points, writing force equations with correct signs, applying energy conservation, and understanding the minimum speed condition—every problem follows the same pattern. Try the vertical circular motion worksheet on this site to test your understanding with different scenarios: varying masses, different radii, and both string and track problems. The more you practice identifying which equations to use at which points, the more automatic the process becomes. Remember that tension at the bottom is always greater than at the top, that minimum speed depends only on radius, and that energy conservation is your bridge between different points in the loop.