Find the speed of a pendulum bob at the bottom of its swing if its initial displacement is 5 degrees and its period is 2 s.

The speed at the bottom of the swing is the maximum value. It is equal to (linear amplitude)*(angular velocity)

= Vmax = L*sin5*(2*pi*f)

You are going to need the length of the pendulum, L, to get the amplitude.

f = 0.5 Hz = (1/(2 pi))*sqrt(g/L)
sqrt(g/L) = 3.14
g/L = 9.86 s^-1
L = 0.994 m

Vmax = 0.27 m/s

Thanks, I couldn't find an equation that derives the max velocity of the pendulum in my textbook.

To find the speed of a pendulum bob at the bottom of its swing, you can use the equation for the period of a simple pendulum and the relationship between speed and distance traveled.

The equation for the period of a simple pendulum is given by:

T = 2π * sqrt(L/g)

where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

In this case, we know the period is 2 seconds, but we need to find the length of the pendulum to calculate the speed at the bottom. However, the initial displacement is given as 5 degrees. To find the length of the pendulum, we need to convert the angle to radians.

1 radian = 180/π degrees

So, 5 degrees is equal to 5 * (π/180) radians.

Now we can substitute the known values into the equation:

2 = 2π * sqrt(L/g)

The g value is approximately 9.8 m/s^2. Rearranging the equation to solve for L:

L = (2/π)^2 * g * T^2

L = (2/π)^2 * 9.8 * 2^2

L = 2.006 m

Now that we know the length of the pendulum is 2.006 meters, we can calculate the speed at the bottom of the swing.

At the bottom of the swing, the pendulum bob is at its lowest point and has its maximum kinetic energy. The total mechanical energy of the pendulum is conserved, so the potential energy at the top is converted to kinetic energy at the bottom.

The maximum potential energy of a pendulum is given by:

PE = m * g * h

where m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the height.

At the top of the swing, the height is equal to the length of the pendulum (h = L). At the bottom of the swing, the height is equal to zero (h = 0). Therefore, the total change in height is L - 0 = L.

The maximum kinetic energy is given by:

KE = PE = m * g * L

The speed of the pendulum bob at the bottom of the swing is given by:

v = sqrt(2 * KE / m)

Substituting the values:

v = sqrt(2 * (m * g * L) / m)

v = sqrt(2 * g * L)

v = sqrt(2 * 9.8 * 2.006)

v ≈ 6.33 m/s

Therefore, the speed of the pendulum bob at the bottom of its swing is approximately 6.33 m/s.