# Centre of Mass: Motion in a Vertical Circle

### Definition and Key Concepts

• Motion in a vertical circle refers to an object moving along a circular path in a vertical plane.
• This differs from uniform motion in a circle because of the effect of gravity, which causes the speed of the object to vary.
• At the highest and lowest points of the circle, the forces acting on the object are at their extremes. At the top, they consist of the tension in the string and the object’s weight; at the bottom they act in opposite directions.
• Tension forces and gravitational forces are key elements to consider in this motion.

### Gravitational Force

• The gravitational force acting upon the object, commonly referred to as the object’s weight, is directed towards the centre of the Earth, or downwards.
• When moving in a vertical circle, the effects of gravity cause the object’s speed and kinetic energy to fluctuate.

### Tension Force

• The tension force in the string (or rod) connects the moving object to the centre of the circular path.
• The magnitude of this force varies depending upon the object’s position within the circle. It reaches its maximum value at the bottom of the circle and is at a minimum at the top.
• At the very top of the circle, the tension in the string must be at least equal to the weight of the body to prevent it from falling.

### Essential Principles

• The object moving in a vertical circle has both kinetic and potential energy.
• As the object moves up, it slows down due to the increase in gravitational potential energy. Conversely, as it descends, its speed increases due to the loss of potential energy and gain in kinetic energy.
• The conservation of mechanical energy can be applied within this context: assuming no air resistance, the sum of kinetic and potential energy remains constant throughout the motion.

### Practical Applications

• Understanding motion in a vertical circle has direct practical implications, for instance, in the designing of theme park rides or pendulum clocks.
• From a safety perspective, grasping these principles is key in industries which involve movement in a vertical plane - e.g., construction cranes.

### Problem-Solving involving Motion in a Vertical Circle

• Problems in this area typically involve finding the minimum speed at the top of the circle for the object to maintain circular motion, or the variations in tension in the string at different points.
• In solving these problems, ensure you separate the forces acting vertically and those acting towards the centre of the circle.
• Another common task in problem-solving is applying the principle of conservation of energy.

Understanding the intricacies of motion in a vertical circle is not only fascinating but also imperative for practical applications ranging from everyday objects, such as pendulums, to adventurous amusement park rides.