A pendulum consist of sphere of mass m, attached to light cord of length L, The sphere is released from rest when a cord makes an angle Q, with vertical and pivot at p is frictionless. (a)find the speed of the sphere when it is at lowest point.(b) what is tension in the cord B.

To tackle this problem involving a pendulum, we need to analyze the motion of the sphere as it swings down from its initial position to the lowest point. We'll use principles from physics, specifically energy conservation and dynamics, to find both the speed of the sphere at the lowest point and the tension in the cord at that moment.

Finding the Speed of the Sphere at the Lowest Point

When the pendulum is released from an angle θ with the vertical, it possesses gravitational potential energy due to its height above the lowest point. As it swings down, this potential energy is converted into kinetic energy. We can use the conservation of mechanical energy to find the speed.

Step 1: Calculate the Initial Height

The height (h) of the sphere when it is at angle θ can be determined using trigonometry. The vertical height from the lowest point to the initial position is given by:

• h = L - L cos(θ) = L(1 - cos(θ))

Here, L is the length of the cord, and cos(θ) gives us the vertical component of the length when the pendulum is at angle θ.

Step 2: Apply Conservation of Energy

The total mechanical energy at the top (potential energy) is equal to the total mechanical energy at the bottom (kinetic energy). Thus, we have:

• Potential Energy at the top: PE = mgh = mgL(1 - cos(θ))
• Kinetic Energy at the bottom: KE = (1/2)mv²

Setting these equal gives us:

• mgL(1 - cos(θ)) = (1/2)mv²

We can cancel the mass (m) from both sides since it appears in both terms:

• gL(1 - cos(θ)) = (1/2)v²

Step 3: Solve for Speed (v)

Rearranging the equation to solve for v, we get:

• v² = 2gL(1 - cos(θ))
• v = √(2gL(1 - cos(θ)))

This equation gives us the speed of the sphere at the lowest point of its swing.

Determining the Tension in the Cord at the Lowest Point

Next, we need to find the tension in the cord when the sphere is at the lowest point. At this position, two forces act on the sphere: the gravitational force (mg) acting downward and the tension (T) in the cord acting upward.

Step 1: Analyze Forces at the Lowest Point

At the lowest point, the sphere is in circular motion, which means there is a net centripetal force directed towards the pivot. The net force can be expressed as:

• Net Force = T - mg = m(v²/L)

Step 2: Rearranging for Tension (T)

We can rearrange this equation to solve for the tension:

• T = mg + m(v²/L)

Step 3: Substitute for v²

Now, substituting our expression for v² from earlier:

• T = mg + m(2g(1 - cos(θ)))

Factoring out m gives us:

• T = m(g + 2g(1 - cos(θ)))
• T = mg(3 - 2cos(θ))

This equation provides the tension in the cord when the sphere is at the lowest point of its swing.

Summary

In summary, the speed of the sphere at the lowest point is given by v = √(2gL(1 - cos(θ))), and the tension in the cord at that point is T = mg(3 - 2cos(θ)). These results illustrate the interplay between gravitational potential energy and kinetic energy, as well as the forces acting on an object in circular motion.