Energy of pendulum physics

Question

A wrecking ball on a 15.4 m long

cable is pulled back to an angle of

33.5° and released. At what speed

is it moving at the bottom of its

swing?

-I need to know how fast it would be going at the bottom of the swing ,Plzz I need a direct answer ;I need to check my work …!I would so greatly appreciate it

Answer 1

Final Answer: 7.1 m/s

Explanation: A pendulum is a neat example of kinetic energy and gravitational potential energy constantly being converted back and forth. As it moves towards the bottom more and more potential energy is converted into kinetic, and as it moves up that kinetic energy is converted back into gravitational potential.

At the bottom of its swing all of the gravitational potential energy will have been converted into kinetic energy so we can write:

U_i = K_f (Initial Potential energy = Final kinetic energy)

U_i = mgh (mass x gravity x height)

K_f = 1/2 mv_f² ( 1/2 x mass x velocity final squared)

This gives us

mgh = 1/2 mv_f²

As you can see we can cancel out the mass, so the mass of the pendulum isn't important and is why it's often not included in these kinds of questions. Then after multiplying by 2 and taking the square root of both sides to isolate velocity we have

(2gh)^1/2 = v_f

Now we simply solve for the height difference at the lowest point and we can plug in the rest of the known values.

At the bottom of the swing the ball will have a vertical distance from the fulcrum equal to the full length of the cable. At the top of the swing, that distance will be the cable length * cos(θ). So our height at any given point will be:

L - L cos(θ)

We can thus now write:

(2g(L - L cos(θ)))^1/2 = v_f

(2 * 9.81 * (15.4 - 15.4 cos(33.5)))^1/2 = V_f

7.1 = V_f

So our final velocity will be 7.1 m/s