the bob of a simple pendulum moves simple harmonically with amplitude 8.0cm and perid 2.00s. its mass is o.50kg. the motion of the bob is undemped. calculate the maximum values for:-
{a} the speed of the bob
{b} the kinetic energy of the bob.
say
x = A sin (2 pi t/T) = 0.08 (sin pi t)
then
v = 0.08 pi cos (pi t)
so max v = 0.08 pi
and (1/2) m v^2 = (1/2)(0.50) pi^2 (0.0064)
God punish you
To calculate the maximum values for the speed and kinetic energy of the bob in a simple pendulum, you can use the formulas for simple harmonic motion:
The formula for the period of a simple pendulum is given by:
T = 2π√(L / g)
Where:
T = Period of the pendulum
L = Length of the pendulum
g = Acceleration due to gravity (approximately 9.8 m/s²)
Given:
Amplitude (A) = 8.0 cm = 0.08 m
Period (T) = 2.00 s
Mass (m) = 0.50 kg
Step 1: Determine the length of the pendulum
Using the formula for the period of a pendulum, we can rearrange it to solve for the length (L):
T = 2π√(L / g)
Rearranging the formula:
(L / g) = T² / (4π²)
L = (T²g) / (4π²)
Plugging in the given values:
L = (2.00² * 9.8) / (4π²) ≈ 0.510 m
Step 2: Calculate the maximum speed (v) of the bob
The maximum speed occurs when the pendulum passes through the equilibrium position. The maximum speed (vmax) is given by:
vmax = A * ω
Where:
A = Amplitude of the motion
ω = Angular frequency
The angular frequency can be calculated using:
ω = 2π / T
Plugging in the given values:
ω = 2π / 2.00 ≈ 3.14 rad/s
vmax = 0.08 * 3.14 ≈ 0.2512 m/s
So, the maximum speed of the bob is approximately 0.2512 m/s.
Step 3: Calculate the maximum kinetic energy (KE) of the bob
The kinetic energy of an object is given by:
KE = (1/2) * m * v²
Where:
m = Mass of the object
v = Velocity of the object
Plugging in the given values:
KE = (1/2) * 0.50 * (0.2512)² ≈ 0.0158 J
So, the maximum kinetic energy of the bob is approximately 0.0158 Joules.