object falling from pendulum?

Question

a pendulum is released from rest at a displacement of .28 m from its equilibrium position. it is stopped abruptly and uniformly at its equilibrium position and it is observedthat a loose bit of metal slides without resistance of the pendulum and falls to the floor 2.06 m below.

if the projectile started off with a velocity in just its horizontal direction, and if travels .44 m in the horizontal direction during its fall, what was the velocity of the pendulum at equilibrium?

what is the period of motion of the pendulum?

Answer 1

In a time T it travels 0.44 m horizontally.
So hor speed = 0.44 / T m/s

In time T it falls a vertical height of 2.06 m with zero initial vertical velocity
s = g/2 * T^2 = 2.06
T = 0.648 sec
v = 0.44 / 0.648 = 0.679 m/s

The angular frequency of a pendulum is sqrt(g/L)
θ = A*sinωt
dθ/dt = Aω*cosωt
So the max angular velocity = Aω
Max speed of pendulum = Aω*L
We can find A from the initial displacement, which should be an angular displacement, but you said 0.28 m. How is that length measured? Vertically, arc length?

Let's suppose you really means 0.28 radians.
Then A = 0.28
Now Aω*L = 0.679
0.28 * sqrt(g/L) * L = 0.679
sqrt(gL) = 2.425
L = 0.6 m

Now the angular frequency = sqrt(g/L) = 4.046 rad/sec
= 2π / Tp
Tp = 1.55 seconds
This is the period of the pendulum

Answer 2

the first part is found by looking at
vh*t=.44
and
2.06=.5*g*t^2
t=sqrt(2.06*2/g)
so
vh=.44/sqrt(2.06*2/g)
using g=9.81
vh=0.679 m/s

Since there was no resistance as the bit of metal slid off, it can be assumed to be an independent body moving at the same speed as the pendulum at the instant it began to decelerate.

I will have to come back to the period since I have an appointment.

First I will check to make sure the problem is consistent since you didn't have to provide the vertical displacement during the fall. The speed could be calculated using conservation of energy as
.5*m*v^2=m*g*h
or
v=sqrt(2*g*.28)
or 2.34 m/s, which is not consistent with the dynamics of the piece that was ejected.

Could it be that the .28m was on the run of the arc?
Then, knowing that the velocity of the pendulum at the equilibrium point you can calculate the angular velocity of the pendulum as
v/r=w
since we want to solve for r, find the vertical displacement of the pendulum to achieve the vh
m*g*h=.5*m*v^2
h=.5*v^2/g

actually, we need to know the angle of displacement, and there isn't sufficient info to calculate it. That would lead to the r, which would be used to calculate the period.

j

Answer 3

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