Review of the Basic Principles of Circular Motion

Before diving into specific scenarios, let’s review some key concepts:

In the case of a mass on a string, the tension in the string provides the centripetal force necessary for circular motion.

Horizontal Circular Motion

Consider a mass attached to a string, swinging in a horizontal circle.

Force Analysis

In this scenario, two main forces act on the mass:

For purely horizontal motion, the tension must be significantly greater than the weight. The centripetal force is provided entirely by the tension:

$$F_c = T = \frac{mv^2}{r}$$

Where:

What are the key Relationships in Circular Motion?

Conical Pendulum: String at an Angle

When the string makes an angle with the vertical, we have a conical pendulum.

Force Analysis

In this case, we need to consider the components of the tension:

Key Equations

$$\frac{mv^2}{r} = T\sin\theta$$ $$mg = Tcos θ$$ $$\frac{v^2}{rg} = \tan\theta$$

From this, we can derive an expression for velocity:

$$v = \sqrt{rg\tan\theta}$$

Vertical Circular Motion

When a mass on a string is swung in a vertical circle, the analysis becomes more complex due to the changing direction of forces.

Force Analysis

The tension and net force vary depending on the position:

$$T_t = \frac{mv^2}{r} - mg$$ $$T_b = \frac{mv^2}{r} + mg$$

Minimum Velocity

For the mass to complete a vertical circle, there’s a minimum velocity required at the top:

$$v_{min} = \sqrt{gr}$$

If the velocity is less than this, the mass will fall before completing the circle.

Key Takeaways