A pendulum of length L = 48.5 cm and mass m = 169 g is released from rest

when the cord makes an angle of 65.4� with the vertical. What is the speed (m/s) of the
mass upon reaching its lowest point?

To find the speed of the mass when it reaches its lowest point, we can use the principle of conservation of mechanical energy. The total mechanical energy of the mass is constant and is equal to the sum of its potential energy and kinetic energy.

First, let's calculate the potential energy of the mass at its initial position when the cord is making an angle of 65.4 degrees with the vertical. The potential energy (PE) is given by the formula:

PE = m * g * h

where m is the mass of the pendulum, g is the acceleration due to gravity, and h is the height of the mass above its lowest point. Since the mass is released from rest, the initial velocity is zero, so the entire potential energy is converted into kinetic energy at the lowest point of the swing.

Next, let's calculate the height (h) of the mass above its lowest point. The height is given by the formula:

h = L - L * cos(theta)

where L is the length of the pendulum and theta is the angle the cord makes with the vertical. Substituting the given values:

L = 48.5 cm = 0.485 m
theta = 65.4 degrees

h = 0.485 m - 0.485 m * cos(65.4 degrees)

Now, we can calculate the potential energy (PE) at the highest point and convert it into kinetic energy (KE) at the lowest point:

PE = m * g * h
KE = 0.5 * m * v^2

where v is the velocity of the mass at the lowest point.

Setting the potential energy at the highest point equal to the kinetic energy at the lowest point:

m * g * h = 0.5 * m * v^2

Now, we can solve for v:

v^2 = 2 * g * h
v = sqrt(2 * g * h)

Substituting the given values:

v = sqrt(2 * 9.8 m/s^2 * (0.485 m - 0.485 m * cos(65.4 degrees)))

Now, we can calculate the value of v:

v ≈ 1.651 m/s

Therefore, the speed of the mass upon reaching its lowest point is approximately 1.651 m/s.