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Can you please explain how the solved the intergral of "VA"

To explain the solution for the integral of \( V_A \), let's break down each step in detail.

### Step-by-Step Explanation:

1. **Expression for \( V_A \)**:
   The given expression for the potential energy \( V_A \) is:
   \[
   V_A = \int_{\vec{r}_B}^{\vec{r}_A} \vec{F} \cdot d\vec{r}.
   \]
   This integral calculates the work done by the forces acting on the mass as it moves from point \( B \) to point \( A \).

2. **Setting up the forces**:
   The forces acting on the particle include the tension \( \vec{T} \) in the string and the weight \( \vec{W} \) of the mass. Here, we focus on the component of the weight because tension does no work (it's always perpendicular to the displacement). 

   The weight can be expressed as:
   \[
   \vec{W} = mg (\cos(\theta) \hat{r} + \sin(\theta) \hat{\theta}),
   \]
   where \( m \) is the mass of the particle, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle.

3. **Parameterization of the path**:
   The path of the mass is along a circular arc, so we parameterize the position vector \( \vec{r} \) in terms of the angle \( \theta \), such that:
   \[
   d\vec{r} = \ell d\theta (\cos(\theta) \hat{x} + \sin(\theta) \hat{y}),
   \]
   where \( \ell \) is the length of the string.

4. **Dot product calculation**:
   The integral simplifies to:
   \[
   V_A = -mg\ell \int_0^\pi \sin(\theta) d\theta.
   \]
   This expression is derived because we take the dot product of the weight vector and the displacement vector along the path. The negative sign indicates work done against gravity.

5. **Solving the integral**:
   To solve:
   \[
   \int_0^\pi \sin(\theta) d\theta = [-\cos(\theta)]_0^\pi = -\cos(\pi) + \cos(0) = -(-1) + 1 = 2.
   \]
   Hence, substituting back gives:
   \[
   V_A = -mg\ell \cdot 2 = -2mg\ell.
   \]
   
Thus, the change in potential energy from \( B \) to \( A \) is \( -2mg\ell \), indicating the amount of work required to move the mass from point \( B \) to point \( A \).

Would you like more details on any specific step, or have any questions?

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Here are 5 related questions to expand on this topic:

1. How would the calculation change if the string length was variable during the motion?
2. What would be the effect of including air resistance on the integral calculation?
3. Can you derive the speed of the mass at point \( A \) using conservation of energy principles?
4. How would the forces change if the mass was swinging on a parabolic trajectory instead of a circular path?
5. What would be the condition for the mass to maintain a uniform circular motion?

**Tip:** When evaluating definite integrals of trigonometric functions, remember to use the fundamental identities and pay attention to the limits of integration for accurate results.

Can you please explain how they solved the integral of 'V_A'?

Understand how to solve the integral of potential energy (V_A) in a circular motion system where a mass m swings on a string of length ℓ. Learn how to set up the integral of gravitational force in terms of angle θ, apply the work-energy principle, and solve for V_A as the particle moves along the path. Suitable for undergraduate physics students.

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