A pendulum consist of a particle P of mass m, attached to one end of a model string of length l=2.85m. The other end of the string is attached to a fixed point O and the particle moves in a vertical circle about the fixed point.

The angle made by the string with the vertical is measured anticlockwise and we assume that the only forces acting on the particle are its weight W and the tension force T due the string. The particle starts at the lowest point with an initial horizontal velocity of sqrt(7gl/2) in the clockwise direction.

Calculate the speed of the particle, in m/s when the tension becomes zero.

To calculate the speed of the particle when the tension becomes zero, we need to consider the conservation of mechanical energy.

At the lowest point, the particle has gravitational potential energy and kinetic energy. As it moves up the circular path, the gravitational potential energy is converted into kinetic energy. At the highest point, all the potential energy is converted into kinetic energy. At this point, the tension in the string becomes zero.

Let's denote the speed of the particle at the lowest point as v₀, and the speed at the highest point as v.

Using the conservation of energy, we can equate the initial mechanical energy (at the lowest point) to the final mechanical energy (at the highest point):

Initial energy = Final energy

Let's calculate the initial energy at the lowest point:
Initial energy = Potential energy + Kinetic energy

Potential energy at the lowest point is zero, as it is chosen as the reference point.
Kinetic energy at the lowest point = (1/2) * m * v₀²

Now let's calculate the final energy at the highest point:
Final energy = Potential energy + Kinetic energy

Potential energy at the highest point = mgh, where h is the height above the reference point (the lowest point in this case). At the highest point, the height is equal to 2l (the length of the string).
Potential energy at the highest point = m * g * 2l

Kinetic energy at the highest point = (1/2) * m * v²

Equating the initial energy and the final energy:

(1/2) * m * v₀² = m * g * 2l + (1/2) * m * v²

Canceling out the m factor on both sides:

(1/2) * v₀² = g * 2l + (1/2) * v²

Multiplying both sides by 2:

v₀² = 2g * 2l + v²

Simplifying:

v₀² - v² = 4gl

Rearranging the equation:

v² = v₀² - 4gl

Substituting the given value for v₀ = sqrt(7gl/2):

v² = (sqrt(7gl/2))² - 4gl

v² = (7gl/2) - 4gl

v² = 7gl/2 - 8gl/2

v² = -gl/2

Since the square of a speed cannot be negative, we can conclude that the speed of the particle when the tension becomes zero is 0 m/s.