Physics OnlineBallistic Pendulum

Introduction

In this experiment, we will use a ballistic pendulum to calculate the speed of a projectile. We will compare this speed to a theoretical value to test the laws of conservation of energy and momentum.

A ballistic pendulum consists of an object suspended vertically from a support by a rod or cable. You can fire a projectile horizontally into the pendulum, causing it to swing. The maximum height attained by the pendulum can then be measured. This height can be used to calculate the initial speed of the projectile

There are two parts to this calculation. First the ball hits and sticks to the pendulum. This is an inelastic collision, by definition. Momentum will be conserved during the collision because external forces are negligible. The mathematical relationship is as follows:

p = momentum, v = velocity, m = mass, 1 = just before collision, 2 = just after collision, b = ball, p = pendulum

[1] Σp1 = Σp2 (conservation of momentum for a collision of any type)

[2] mbvb1 + mpvp1 = mbvb2 + mpvp2 (using the definition of momentum in equation [1])

[3] vp1 = 0 (since the pendulum begins at rest)

[4] vb2 = vp2 = v2 (since the ball and pendulum stick together)

[5] mbvb1 = (mb+ mp)v2 (substitution of equations [4] and [3] in equation [2])

[6] vb1 = (1+ mp/mb)v2 (equation [5] solved for the variable of interest)

The masses on the right hand side can be easily measured with a scale, but the velocity of the group just after the collision is difficult to directly measure. Instead, you can use the final height of the ball and pendulum to compute the velocity of the group. This is done with an energy calculation for the ball and pendulum system after the collision. There are two forces acting on the pendulum and ball system. There is a variable tension force that does no work (perpendicular force), and a gravitational force by the earth that can be taken into account by including the earth as part of the system and using gravitational potential energy as follows:

2 = just after collision, 3 = pendulum at maximum height

[7] U2 + K2 + Wnc = U3 + K3 (conservation of energy)

[8] U2 = (mb+ mp)gy2 (gravitational potential energy)

[9] K2 = ½(mb+ mp)v22 (definition of kinetic energy)

[10] Wnc = 0 (since no significant non-conservative forces act on the system)

[11] U3 = (mb+ mp)gy3 (gravitational potential energy)

[12] K3 = ½*mv32 = 0 (since the ball and pendulum reach maximum height)

[13] (mb+ mp)gy2 + ½*(mb+ mp)v22 + 0 = mgy3 + 0 ([8] through [12] in [7])

[14] v2 = sqrt[2g(y3 – y2)] (equation [13] solved for v2, the speed of the group)

Equation [14] includes the change in height which cannot be directly measured. We can use trigonometry to relate the length of the pendulum and the maximum angle, both of which we can measure, to the change in height. If we call the length of the pendulum “L”, then the initial vertical distance from the pivot point to the pendulum will be L. After the pendulum swings up, the vertical distance from the pivot point will be L*cosθ. The difference between these two gives the change in height.

[15] y3 – y2 = L – L*cosθ

[16] y3 – y2 = L(1 – cosθ)

Equation [16] can be substituted into equation [14]. Then equation [14] can be substituted into equation [6] to obtain a single equation for the speed of the projectile in terms of known or easily measured physical quantities. This is left as an exercise for the student. Physics is fun!

Equipment You Procure

  • digital camera
  • metric tape measure

Equipment from Kits

  • ballistic pendulum
  • bubble level
  • digital scale

Experimental Procedures

1)Derive a symbolic formula for the initial speed of the ball in terms of measurable physical quantities.

2)Measure the mass of the projectile (it should be around 8 g) and the mass of the pendulum.

3)Level and test fire the ballistic pendulum. You may need to adjust the length of one or more strings to make the ball remain in the pendulum.

4)Measure the vertical distance from the pivot point (top end of a string) of the pendulum to the center of the pendulum.

5)Use a consistent notch in the spring gun and fire the projectile into the pendulum five times. The ball must remain in the pendulum after impact. Record the angle attained by the pendulum in each trial.

6)Average the five measurements of angle.

7)Calculate the experimental initial speed (and error) of the projectile.

8)Compare your experimental speed to the speed provided by the manufacturer of the ballistic pendulum. (notch 1: v = 5.40 ± 0.25 m/s, notch 2: v = 6.40 ± 0.25 m/s, and notch 3: v = 7.70 ± 0.30 m/s)

9)Repeat steps 5 through 8 with a different notch.