a pendulum has a bob of 0.20kg and is suspended by a string of negligible mass near the earth's surface. the pendulum is displaced from its equilibrium position by the angle of 8.0 degrees. it passes the equilibrium position 0.50 s after being released. calculate the length of the pendulum,max. kinetic energy and max. velocity of the bob!

Get the length L from the period P.

The period is 2 seconds because it takes 1/2 second to accomplish 1/4 of it. (maximum deflection to zero deflection)

P = 2 pi sqrt(L/g)= 2 seconds

L/g = (1/pi)^2
L = 0.994 m

The maximum height that it gets raised is
h = L (1 - cos 8) = 0.00967 m

The maximum K.E. is M g h (which is the maximum potential energy)

Use that to compute Vmax

To solve this problem, we can use the equations of motion for simple harmonic motion (SHM). Let's go step by step to find the length of the pendulum, maximum kinetic energy, and maximum velocity of the bob.

Step 1: Finding the period of the pendulum
The period of a pendulum is given by the formula:
T = 2π√(L/g)
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity (approximately 9.8 m/s²).

From the given information, we know that the time taken to reach the equilibrium position is 0.50 seconds. This is half of the period, so we can rewrite the equation as:
0.50 = (1/2)T
Therefore, T = 1 second.
Substituting this into the equation, we have:
1 = 2π√(L/9.8)
Simplifying, we get:
1 = √(315.68/L)
Squaring both sides, we get:
1 = 315.68/L
Rearranging, we find:
L = 315.68 meters.

Therefore, the length of the pendulum is 315.68 meters.

Step 2: Finding the maximum kinetic energy
The maximum kinetic energy (KEmax) of a pendulum occurs when it passes through the equilibrium position. At this point, all of the potential energy is converted into kinetic energy. The potential energy at the maximum displacement is given by:
PE = mgh
where m is the mass of the bob, g is the acceleration due to gravity, and h is the height. As the height is proportional to the square of the amplitude, we can write:
h = (L/2)(1-cosθ)
where θ is the angle of displacement.

Substituting the values, we get:
h = (L/2)(1-cos8°)
h = (315.68/2)(1-cos8°)
h = 0.0866 meters

Now we can calculate the potential energy:
PE = mgh
PE = 0.20 * 9.8 * 0.0866
PE = 0.169 Joules

Since all the potential energy is converted to kinetic energy at the maximum displacement, the maximum kinetic energy (KEmax) is also 0.169 Joules.

Step 3: Finding the maximum velocity
The maximum velocity of the bob occurs when it passes through the equilibrium position. It can be calculated using the equation:
vmax = ωA
where vmax is the maximum velocity, ω is the angular frequency, and A is the amplitude.

The angular frequency (ω) can be calculated using the formula:
ω = 2π/T
where T is the period of the pendulum.

Substituting the values, we get:
ω = 2π/1
ω = 2π rad/s

Now, we can calculate the maximum velocity:
vmax = ωA
vmax = 2π * (L/2)sinθ
vmax = 2π * (315.68/2)sin8°
vmax = 2.79 m/s

Therefore, the maximum velocity of the bob is 2.79 m/s.