A pendulum consists of an object of mass m = 1.2 kg swinging on a massless string of length l = 259 cm. The object has a speed of 2.4 m/s when it passes through its lowest point. What is the greatest angle with the vertical that the string makes during the motion of the object?

Thank you

This is geometry. Draw the figure.

Set mgh=1/2 mv^2, solve for h. That is the height to which the bob rises.

Draw the triangle. If I remember (in my head), Costheta=(l-h)/l
and you solve for the angle.

check that carefully.

To find the greatest angle with the vertical that the string makes during the motion of the object, we can use the conservation of mechanical energy for a pendulum.

The mechanical energy of a pendulum is given by the sum of its kinetic energy and potential energy. At the lowest point of the pendulum's motion, all of the object's energy is in the form of kinetic energy. As the pendulum swings upwards, the kinetic energy is gradually converted into potential energy. At the highest point of the motion, all of the energy is in the form of potential energy.

Mathematically, the conservation of mechanical energy can be expressed as:

K_initial + U_initial = K_final + U_final

where:
K_initial is the initial kinetic energy
U_initial is the initial potential energy
K_final is the final kinetic energy
U_final is the final potential energy

In this case, when the object passes through its lowest point, its speed is given as 2.4 m/s. We can find the initial kinetic energy using the formula:

K_initial = (1/2) * m * v^2

where:
m is the mass of the object
v is the velocity of the object

Using the given values, we can calculate the initial kinetic energy:

K_initial = (1/2) * 1.2 kg * (2.4 m/s)^2

Next, we need to find the final potential energy. The potential energy of a pendulum is given by the formula:

U = m * g * h

where:
m is the mass of the object
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the height of the object above the reference point (in this case, the lowest point of the motion)

At the highest point of the pendulum's motion, its height above the reference point is equal to the length of the string. Therefore:

h = l

We can now calculate the final potential energy:

U_final = m * g * l

Finally, we can substitute the initial kinetic energy and final potential energy into the conservation of mechanical energy equation:

K_initial + U_initial = K_final + U_final

Since all the initial energy is in the form of kinetic energy and all the final energy is in the form of potential energy, we can simplify the equation to:

K_initial = U_final

Solve for the angle:

U_final = m * g * l
K_initial = (1/2) * m * v^2

Setting them equal to each other:

(1/2) * m * v^2 = m * g * l

Simplifying and solving for the angle:

theta = arcsin((v^2)/(2 * g * l))

Plug in the values:

theta = arcsin((2.4^2)/(2 * 9.8 * 2.59))

Use a scientific calculator to find the arcsin value, giving you the greatest angle with the vertical that the string makes during the motion of the object.