a simple pendulum of lengh l,the mass of whose bob is m,is obseved to have a speed v when the cord makes an angle with the vertical.Determine the least value v2 that v could have if the cord is to achieve a horizontal position during the motion

To determine the least value of v² that v could have if the cord is to achieve a horizontal position during the motion, we need to analyze the energy of the system.

The total mechanical energy of a simple pendulum is given by the sum of its kinetic energy (KE) and its potential energy (PE). The kinetic energy depends on the speed (v) of the bob, and the potential energy depends on the height (h) of the bob.

At the highest point of its swing, when the cord makes an angle θ with the vertical, the bob is momentarily at rest, so its speed is zero (v = 0). At this point, all the initial potential energy has been converted to kinetic energy.

Hence, we can equate the potential energy at the highest point to the kinetic energy at the lowest point when the cord is horizontal:

mgh = mv²/2

Here,
m = mass of the bob,
g = acceleration due to gravity (approximately 9.8 m/s²),
h = height from the lowest point to the highest point (considering the length of the pendulum l and the angle θ).

To determine the least value of v², we need to minimize the height h. The minimum height will occur when the angle θ is maximum, i.e., when θ = 90 degrees.

In a simple pendulum, the height (h) can be calculated as follows:

h = l - l * cos(θ)

Now, substituting θ = 90 degrees and simplifying the equation, we get:

h = l - l * cos(90 degrees)
h = l - l * 0
h = l

Substituting this value of h into the energy equation:

mgh = mv²/2
m * g * l = m * v²/2
v² = 2 * g * l

Therefore, the least value of v² that v can have to achieve a horizontal position during the motion is 2 * g * l, where g is the acceleration due to gravity and l is the length of the pendulum.