Question

In a conical pendulum, a bob moves in a horizontal circle, as shown in the figure. The period of the pendulum (the time it takes for the bob to perform a complete revolution) is a) \(T=2 \pi \sqrt{L \cos \theta / g}\). b) \(T=2 \pi \sqrt{g \cos \theta / L}\). c) \(T=2 \pi \sqrt{L g \sin \theta}\). d) \(T=2 \pi \sqrt{L \sin \theta / g}\). e) \(T=2 \pi \sqrt{L / g}\).

Short answer

Answer: The correct expression for the period of a conical pendulum is \(T = 2 \pi \sqrt{L \sin \theta / g}\).

Step-by-step solution

Analyze Forces in Vertical Direction

In the vertical equilibrium, the tension from the string (Tsinθ) should counterbalance the gravitational force (mg). So, we can write that \(T \sin \theta = m g\) where 'm' is the mass of the bob.

Analyze Forces in the Horizontal Direction

In the horizontal direction, the centripetal force required for circular motion is provided by the horizontal component of the tension force in the string (Tcosθ). We can write the centripetal force as \(m v^2 / r\), and tension as \(T \cos \theta\). So, \(T \cos \theta = m v^2 / r\) where 'v' is the linear velocity of the bob and 'r' is the radius of the circle.

Relate Velocity to Period

We can find the perimeter of the circle by using the radius 'r' and relate it to the linear velocity 'v' and period 'T'. \(v = 2 \pi r / T\)

Eliminate 'v' and Solve for Period

Now, we can substitute for 'v' in the horizontal equilibrium equation and solve for 'T': \(T \cos \theta = m (2 \pi r / T)^2 / r\) We can also simplify the vertical equilibrium equation as: \(T \sin \theta = m g\) Divide the horizontal equilibrium equation by the vertical equilibrium equation: \(\cos \theta / \sin \theta = (2 \pi r / T)^2 / g\) Rearrange and solve for 'T': \(T = 2 \pi \sqrt{r \sin \theta / g}\) In terms of the string length (L), we can relate the radius 'r' as \(r = L \cos \theta\). Replacing 'r' in the expression for the period: \(T = 2 \pi \sqrt{L \cos \theta \sin \theta / g}\) Comparing this result with options (a) to (e), the expression for the period is closest to option 'd'. Therefore, the correct expression for the period of the conical pendulum is: \(T = 2 \pi \sqrt{L \sin \theta / g}\).

Key concepts

Pendulum Period

The period of a pendulum is the time it takes to complete one full swing, or cycle. In the case of a conical pendulum, this period is the duration required for the pendulum to travel in a complete circle. Understanding the pendulum's period is vital as it reflects the oscillation properties of a system. For a conical pendulum, the period 'T' is affected by the length of the string 'L', the acceleration due to gravity 'g', and the angle 'θ' the string makes with the vertical.

In the step by step solution, we see the period 'T' can be found by balancing the forces in both horizontal and vertical directions, and then by relating the velocity 'v' of the bob to the period. The final expression derived is \(T = 2 \pi \sqrt{L \sin \theta / g}\), which tells us that the period increases with the length of the pendulum and decreases as the gravitational acceleration or the angle 'θ' increases.

Centripetal Force

Centripetal force is the force that keeps an object moving in a circular path and is directed towards the center of the circle. For a conical pendulum, this force is crucial because it maintains the circular motion of the bob. It is provided by the horizontal component of the tension in the string when the bob swings around the central point.

In the exercise solution, we find that the centripetal force, which is represented by the formula \(m v^2 / r\), is equal to the horizontal component of the tension, expressed as \(T \cos \theta\). This balance of forces ensures the bob’s circular trajectory, emphasizing the significance of centripetal force in maintaining circular motion.

Equilibrium of Forces

Equilibrium of forces occurs when all the forces acting on an object are balanced, resulting in no acceleration. The conical pendulum is a perfect illustration of this concept, as it involves balancing the vertical and horizontal forces to maintain its circular motion. The vertical forces consist of the gravitational force and the vertical component of the tension in the string, while the horizontal forces are the centripetal force and the horizontal component of tension.

The step by step solution shows how the forces equilibrate with \(T \sin \theta = m g\) in the vertical direction and \(T \cos \theta = m v^2 / r\) in the horizontal. These equations ultimately allow the calculation of the pendulum period, demonstrating the direct link between equilibrium of forces and the pendulum's motion.

Circular Motion

Circular motion refers to the motion of an object along the perimeter of a circle. In physics, it is important to understand circular motion because it involves acceleration towards the center of the circle even if the speed remains constant. The conical pendulum exhibits uniform circular motion with constant angular speed but changing velocity direction.

The exercise's solution connects circular motion to the concept of centripetal force. The constant change in direction is evidence of a continuous acceleration towards the center, provided by this force. By calculating the period of the motion via the derived formula \(T = 2 \pi \sqrt{L \sin \theta / g}\), we can deepen our understanding of how objects behave when they are in a state of uniform circular motion.