A 1.5 kg frictionless pendulum is released from point A at an angle θ of 35 degrees. The speed of the pendulum at point C is 3.4 m/s. What is the length of the pendulum?

Question

A) 3.3 m

b) 0.72 m

c) 0.59 m

d) 8.7 m

Answer 1

Please try to show the picture like this:

Position 1 of the pendulum at point A, when it makes angle Θ = 35° with the vertical. Position 2 - it is when the pendulum at point C (equilibrium position, and the bob of the pendulum is at a lowest possible level. Label point O - it is the point where we hang up the pendulum; and point B - where vertical is crossed by the horizontal line, that goes through point A.

We can write the law of conservation of mechanical energy

E₁ = E₂

Here E₁ - total mechanical energy at position 1; and E₂ - total mechanical energy of the pendulum at position 2. Let's select the zeroth's level of energy point 2.

Then at point 1 the body has only potential energy E₁ = mgh ( here h = CB). At point 2 the body has only kinetic energy E₂ = 1/2 mv² .

Now we write

mgh = 1/2 mv².

From this equation

h = v² / 2g.

Using geometry, we see that h = CO - BO

CO = L - length of the pendulum.

BO - leg of the right triangle ABO ( right angle is angle ABO), hence

BO = AO·cos Θ or BO = Lcos Θ

Finally, h = L - Lcos Θ = L(1 - cos Θ).

Now we can write

L(1 - cos Θ) = v² / 2g

From this equation we will have for length of the pendulum

L = v² /2g (1 - cos Θ)

Putting numerical values to this formula ( and g = 10 m/s² ), we will have

L = 3.2 m

Now you look for the answer that is the closest to yours. It is (a)!