27-April 2015 Balistic Pendulem

Purpose: To demonstrate the conservation of momentum and energy principle, using the known formulas, and using propagated uncertainty will account for any error in the results.

Theory: The formulas of conservation of momentum (Mass original*Velocity original= Mass final*Velocity final) and the conservation of energy (Kinetic energy initial= Potential energy due to gravity) can be used to find the velocity of a steel ball as it collides into a ballistic pendulum, and any error will be accounted for using propagated uncertainty.

Procedure: Take measurements of the ballistic pendulum apparatus, weigh steel ball, give the ball an initial velocity so it collides with the ballistic pendulum, preform calculations for this initial velocity using the measurements and account for any error.

Apparatus

Measurements

*Steel Ball mass 0.00767 kg +/- 0.0001 kg

*Ballistic Pendulum mass 0.0809 kg +/- 0.0001 kg

*Angle formed from pendulum after collision 0.314 rad +/- 0.00873 rad

*Length of Pendulum 0.02 meters +/-  0.0005 meters

Calculations

* 0.5*M_total*V_final^2= M_total*g*H

*H is calculated using trigonometry

*H=L-L cos (angle)

* Mass' cancel out and

                                                                  0.5*V_f^2=g*H

                                                                   V_f=sqr(2*g*H)

                                                                    V_f=sqr(2*9.8*(0.02-0.02 cos 0.314))

                                                                    V_f= 0.138 m/s

*Mass_ball*V_i=Mass_total*V_f

                                                             V_i=(Mass_total*V_f)/Mass_ball

                                                             V_i=(0.00767+0.0809)*0.138/0.00767

                                                             V_i= 1.59 m/s

*Propagated error

                                                dv= ((M_b+M_p)*sqr(2*g*L-L cos (angle))/M_b

                                               dv/dM_b=sqr(2*g*L-L cos (angle))/(-M_b^2) *dM_b

                                               dv/dM_p=(M_b+1/M_b)*sqr(2*g*L-L cos (angle))*dM_p

                                               dv/dL= ((M_b+M_p)/M_b)*sqr(2*g*1-cos (angle))/2sqr(L) *dL

                                 dv/dang= ((M_b+M_p)/M_b)*sqr(2*g*L))*(sin (angle)/2sqr(1-cos angle) *dang


*Add all the values and dv=+/-0.301 m/s

*Initial velocity is 1.59 m/s +/- 0.301 m/s

Conclusion

*The initial velocity of the steel ball as it collided with the pendulum was 1.59 m/s giver or take 0.301 m/s.