A captive James Bond is strapped to a table beneath a huge
pendulum made of a -diameter uniform circular metal blade
rigidly attached, at its top edge, to a -long, massless rod.
The pendulum is set swinging with a amplitude when its lower
edge is above the prisoner, then the table slowly starts ascending
at . After minutes, the pendulum’s amplitude
has decreased to . Fortunately, the prisoner is freed with a
mere to spare. What was the speed of the lower edge of the
blade as it passed over him for the last time?
The speed of the lower edge of the blade as it passed over him for the last time is and
Data from given information
The length of the pendulumThe radius of blade is
The lower edge is
After minutes, , the amplitude of the pendulum is
To find the time constant of damped oscillation for the pendulum
Calculate the time at which the bond is freed. observe that it takes place before
Then, the velocity in is constant
Bond is saved with to spare, then as we know
Now, we have to determine how many oscillation presented before bond is saved. For that, we have to find the period of oscillation of pendulum
Where is the moment of inertia of system. Therefore, the moment of inertia of system is
Substituting the values in equation
so localid="1650084849770" oscillation before bond is freed.
since, the previous time the blade is above bond would be after localid="1650084855792" oscillations or afterlocalid="1650084863032"
Then, we need to find the speed of the blade at
So, let . Observe that at this time, the energy in the system is entirely kinetic since when the blade is above Bond. Thus, where v is the speed that we want. Further, observe that is entirely gravitational potential since the blade is released from rest. Thus, . Put the two equations in equation and solve for v at
The linear speed of it. The angular speed of the blade can be obtained by
A block attached to a spring with unknown spring constant oscillates with a period of . What is the period if
a. The mass is doubled?
b. The mass is halved?
c. The amplitude is doubled?
d. The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.
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