Core Practical 9: Investigating Impulse

The accelerating force $F$ in this system is provided by the weight of the hanging masses, calculated as $F = mg$, where $m$ is the mass of the hanger and its contents. This force is assumed to be constant throughout the motion between the light gates.

The impulse equation $I = F \Delta t$ can be equated to the change in momentum $\Delta p = M(v_B - v_A)$, where $M$ is the total mass of the system (trolley + all masses). Combining these yields the relationship $mgt = M(v_B - v_A)$.

To linearize this for graphical analysis, the equation is rearranged into the form $y = mx + c$. By plotting $mt$ on the y-axis against $(v_B - v_A)$ on the x-axis, the resulting gradient should be $\frac{M}{g}$ if the relationship holds true.

Friction Compensation: The ramp must be slightly tilted so that the trolley moves at a constant velocity when given a small push without any hanging mass. This ensures that the component of gravity acting down the slope exactly cancels out the frictional forces.

Constant System Mass: It is critical to keep the total mass $M$ of the system constant. This is achieved by starting with all slotted masses on the trolley and moving them one by one to the hanging hanger for each subsequent trial.

Data Acquisition: Light gates are used to measure the instantaneous velocities $v_A$ and $v_B$ as the interrupter card passes through them. The data logger also records the time interval $t$ taken for the trolley to travel from the first gate to the second.

Gradient Interpretation: Always check the axes of the provided graph. If the graph is $mt$ vs $\Delta v$, the gradient is $M/g$; however, if the graph is $\Delta v$ vs $mt$, the gradient becomes $g/M$.

Unit Consistency: Ensure all masses are converted to kilograms (kg) and velocities to meters per second (m/s). A common mistake is using grams (g) for the hanging mass while using kilograms for the trolley mass.

Verification: To verify if the results are reasonable, calculate the theoretical gradient $M/g$ using the measured total mass and compare it to the experimental gradient obtained from the line of best fit.

Ignoring Friction: Failing to tilt the ramp leads to a systematic error where the actual accelerating force is less than $mg$ due to friction. This results in a non-zero intercept on the graph or a curved line.

Interrupter Card Errors: If the width of the card entered into the software does not match the physical width, all velocity readings will be systematically scaled. This affects the magnitude of the change in momentum but may not change the linearity of the graph.

Misaligned Gates: If the light gates are not perpendicular to the trolley's path, the effective length of the interrupter card changes. This introduces random errors in velocity measurements, leading to scatter around the line of best fit.