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SUMMER-1 2012
BALLISTIC PENDULUM
EXPERIMENT 8 – Ballistic Pendulum
OBJECTIVE
The purpose of this experiment is to measure the initial velocity of a projectile in two independent
ways: One by treating it as a projectile moving according to the kinematic equations, and the other
is by applying the conservation of linear momentum and energy to a ballistic pendulum.
MATERIALS
Ballistic Pendulum
Meter stick
Carbon paper
Triple beam balance
White paper
INTRODUCTION
The apparatus used in this experiment is a combination of a ballistic pendulum and a spring gun.
The pendulum consists of a massive cylindrical bob, hollowed out to receive a projectile and
suspended by a strong, light rod pivoted at its upper end. The projectile is a steel ball that is fired
into the pendulum bob and is held there by a spring in such a position that its center of gravity lies
on the axis of the suspension rod. A brass index is attached to the pendulum bob to indicate the
height of the center of gravity of the loaded pendulum. When the projectile is fired into the bob,
the pendulum swings upward and is either caught at its highest point by a pawl that engages a
tooth on a curved rack, or it swings back and leaves an angle indicator at the maximum angle.
Method 1:
If the projectile is fired horizontally with an initial velocity Vo, it will follow the parabolic path
shown in figure 1.The horizontal range X and vertical displacement Y of the projectile are given
by:
X = Vo T
(1)
Y = (1/2) g T 2
(2)
Combining the above equations allows to determine Vo , given by equation ( 3 )
Vo = X •(g/2Y)1/2
(3)
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SUMMER-1 2012
BALLISTIC PENDULUM
Method 2:
The initial velocity Vo of the projectile can also be determined by using the ballistic pendulum (Fig
2). It consists of a spring gun that fires a metallic ball of mass m which is caught by a catcher at
the end of a pendulum of mass M. The collision between the ball and pendulum is perfectly
inelastic. As a result, the combination swings upward until it either stops at the highest point by a
ratchet, or it moves the angle indicator which remains at its highest point as the pendulum swings
back.
In a collision of two bodies, Momentum is always conserved
in the absence of any external force. When the ball impacts
the pendulum, momentum is conserved, which implies that
the momentum of the ball just before it hits the pendulum is
equal to the momentum of the ball + pendulum just after the
collision:
mVo = ( m + M ) V1
(4)
where V1 is the common velocity of pendulum – ball just
after collision.
When the pendulum moves due to the impact, it has a certain kinetic energy, which is converted
into potential energy as the pendulum swings upwards. If the combination rises through a height h,
then from conservation of mechanical energy we have:
(1/2) ( m + M ) V12 = ( m + M ) g h
(5)
By combining ( 4 ) and ( 5 ) we determine the velocity Vo to be:
Vo =
(m+M)
√(2gh)
m
(6)
The height ‘h’ can be found by noting the angle through which the pendulum rises. If the length of
the pendulum (from the pivot to its center of mass) is R, then
h = R (1 – Cos θ )
(7)
The reason why we do not equate the kinetic energy of the ball just before it hits the pendulum to
the potential energy of the ball + pendulum at its highest point is that the collision is inelastic, and
results in a loss of energy (but not of momentum).
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BALLISTIC PENDULUM
EXPERIMENTAL PROCEDURE
Method 1
Projectile Motion
1) Move the pendulum out of the path of the ball, and secure the ballistic pendulum to the table.
2) Place the ball in the shaft in the un-cocked position and measure the height Y from the bottom
of the ball to the ground by using a plumb line.
3) Place the ball in the shaft and cock the gun until the shaft is locked in position. Always use the
ramrod to push the spring in with the ball in place. Fire the gun few times to get an approximate
position of where it strikes the ground, and then tape a piece of white paper and center it around
where the ball lands. You can cover it with a carbon paper if you wish.
4) Shoot the ball at least five times and record the horizontal distance of each mark left on the
paper from the end of the gun barrel.
Method 2
Ballistic pendulum
5) Remove the pendulum from the apparatus and measure its mass M and that of the ball m. Also
measure the length of the pendulum R from the pivot to the center of mass (as indicated on the
pendulum).
6) Reinstall the pendulum.
7) Place the ball in the shaft and cock the gun until the shaft is locked in position. Fire the gun
and after the pendulum swings up, record the angle through which it had moved. Repeat this
procedure five times and record your results.
Note that there are three positions to which you can cock the gun, and there are light and heavy
balls. Use the gun position and ball that gives you a good angular deviation of the pendulum, and a
good range for the ball to hit the floor. For this you would need to shoot the gun a few times. Then
use the selected position and ball for all measurements in this experiment.
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SUMMER-1 2012
BALLISTIC PENDULUM
EXPERIMENT 8 – Ballistic Pendulum
REPORT FORM
Name: _________________________________
Part 1
Date _________________
Projectile Motion
Shots
Range, X
Deviation = X - Xave
1
2
3
4
5
Average, Xave
Vertical distance Y __________
Velocity Vo of projectile from eqn. 3: _____________
Part 2
Ballistic Pendulum
Mass ‘M’ of pendulum _________
Mass ‘m’ of ball
_________
Trials
1
2
3
4
Angle θ
Vertical distance h
Velocity Vo of projectile from ( 6 ) _____________
Percent difference between the two values of Vo
_____________
Kinetic energy of ball + pendulum before collision: _____________
Kinetic energy of ball + pendulum after collision: _____________
Percent loss of kinetic energy: ________________
4
5
Average
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BALLISTIC PENDULUM
CALCULATIONS
1)
Calculate the average value for the range X of the projectile as well as its average deviation.
2)
Using equation ( 3 ), calculate the initial velocity Vo of the projectile.
3)
Calculate the average value for the vertical distance h the pendulum-ball system has risen
after the collision. This is the difference between the height of CM at its highest point and
that at its lowest point.
4)
Using equation ( 6 ), calculate the initial velocity Vo of the projectile.
5)
Calculate the percent difference between the two values of Vo (i.e. by eqn. 3and eqn. 6)
6)
Calculate the kinetic energies before and after collision, and the percent energy lost.
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SUMMER-1 2012
BALLISTIC PENDULUM
EXPERIMENT 8 – Ballistic Pendulum
Pre-Lab Questions
Due before lab begins.
Name _________________________
Date ___________________
1)
If the speed of a particle is doubled, by what factor is its momentum changed? By what
factor is its kinetic energy changed?
2)
One person is standing perfectly still, and then takes a step forward. Is linear momentum
conserved? Explain.
3)
A bullet of mass 12 grams is fired into a large block of mass 1.5 kg suspended from light
vertical wires. The bullet imbeds in the block and the whole system rises 10 cm. Find the
velocity of the bullet just before collision.
4)
Define a perfectly inelastic collision. Give two examples of such a collision.
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SUMMER-1 2012
BALLISTIC PENDULUM
EXPERIMENT 8 – Ballistic Pendulum
Post- Laboratory Questions (Attach with your Report. Answer in the spaces provided)
Name _________________________
Date ___________________
1)
Since all the kinetic energy of the ball just before the collision does not appear as potential
energy of the ball + pendulum, where does the energy ‘lost’ go?
2)
Prove, by making use of equation ( 1 ), that fractional loss of energy is:
3)
Derive Equation ( 6 ).
4)
Derive equation ( 7 ).
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