A 1.5 m long frictionless pendulum of mass 1.6 kg is released from point A at an angle 0 of 10 degrees. What is the speed of the pendulum at point D, half the height up to point E?
To solve this problem, we can use conservation of mechanical energy.
The mechanical energy of the pendulum at point A (initial point) is given by:
E_A = mgh_A + (1/2)mv_A^2
where:
m = mass of the pendulum (1.6 kg)
g = acceleration due to gravity (9.8 m/s^2)
h_A = height of the pendulum at point A (1.5 m in this case)
v_A = velocity of the pendulum at point A
Since the pendulum is released from rest at point A, the initial velocity v_A is zero. Therefore, the mechanical energy at point A simplifies to:
E_A = mgh_A
At point D (final point), the mechanical energy is given by:
E_D = mgh_D + (1/2)mv_D^2
where:
h_D = height of the pendulum at point D (half the height of point A, so h_A/2 = 1.5/2 = 0.75 m)
v_D = velocity of the pendulum at point D (which we want to find)
Conservation of mechanical energy states that the mechanical energy at point A is equal to the mechanical energy at point D:
E_A = E_D
mgh_A = mgh_D + (1/2)mv_D^2
mgh_A - mgh_D = (1/2)mv_D^2
mg(h_A - h_D) = (1/2)mv_D^2
Simplifying further:
g(h_A - h_D) = (1/2)v_D^2
v_D^2 = 2g(h_A - h_D)
Substituting the known values:
v_D^2 = 2(9.8 m/s^2)(1.5 m - 0.75 m)
v_D^2 = 2(9.8 m/s^2)(0.75 m)
v_D^2 = 15.18 m^2/s^2
Taking the square root of both sides to find the speed:
v_D = √(15.18 m^2/s^2)
v_D ≈ 3.89 m/s
Therefore, the speed of the pendulum at point D, half the height up to point E, is approximately 3.89 m/s.