To calculate the kinetic energy at the low point of the pendulum's swing, we need to consider the conservation of mechanical energy. At the highest point of the swing, all of the potential energy is converted into kinetic energy at the lowest point.
The potential energy at the highest point can be calculated using the gravitational potential energy formula:
PE = m * g * h
In this case, the height (h) is given by:
h = L * (1 - cos(θ))
where L is the length of the pendulum and θ is the angle with the vertical (45° in this case).
Now, at the low point of the swing, all of the potential energy is converted into kinetic energy. The kinetic energy (KE) can be calculated using the formula:
KE = (1/2) * m * v²
where v is the velocity of the pendulum at the low point.
To find the velocity, we can use the conservation of mechanical energy:
PE at highest point = KE at lowest point
m * g * h = (1/2) * m * v²
Substituting the values of h and θ:
m * g * L * (1 - cos(θ)) = (1/2) * m * v²
Since mass (m) is common on both sides of the equation, we can cancel it out:
g * L * (1 - cos(θ)) = (1/2) * v²
Now, we can solve for v:
v² = 2 * g * L * (1 - cos(θ))
v = √[2 * g * L * (1 - cos(θ))]
Substituting the given values:
θ = 45° g = 9.8 m/s² L (length of the pendulum) = given value
After calculating v, we can find the kinetic energy (KE) using the formula:
KE = (1/2) * m * v²
Substituting the given mass (m) and the calculated value of v, we can find the kinetic energy at the low point of the pendulum's swing.