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External force impulse changes a system’s total linear momentum, while external torque impulse changes its angular momentum about a specified point.
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Formula
p in kg·m/s; force impulse in N·s; L in kg·m2/s; angular impulse in N·m·s
| Symbol | Meaning | Unit |
|---|---|---|
| Total linear momentum of the system | kg·m/s | |
| Fext | Resultant external force | N |
| Angular momentum about a specified reference point | kg·m2/s | |
| Resultant external torque about that same point | N·m |
\Delta\vec p=\int_{t_1}^{t_2}\vec F_{\mathrm{ext}}dt, \qquad \Delta\vec L=\int_{t_1}^{t_2}\vec\tau_{\mathrm{ext}}dtIntegrating the resultant external force over a time interval gives the change in system momentum. The angular relation similarly integrates external torque about a stated reference point. If the corresponding external resultant is zero, that momentum is conserved. Internal forces cancel in a suitable system balance. For angular momentum, reference-point choice and support reactions matter; zero net force does not automatically imply zero torque about every point.
Specify system, interval and reference point; angular momentum and torque must use the same point.
Time integration of linear and rotational equations gives momentum change.
Average force
Average over the time interval.
A 0.2 kg ball reaches 15 m/s from rest. Find Δp.
Δp=0.2×15=3 N·s.
Counting internal interactions as external force.
State the system boundary; internal forces cancel in pairs.
Formula cluster
Linear and angular impulse change their respective motion quantities.
A 0.20 kg ball speeds from rest to 15 m/s in 0.05 s. Its momentum change is 3.0 N·s, corresponding to a 60 N average resultant force.
Analyze impacts, rotating machinery, angular impulses and system reactions over a time interval.
Here is how to work through a typical Linear and angular impulse–momentum balance (Δp = ∫Fext dt; ΔL = ∫τext dt) task step by step:
Task
A 0.2 kg ball reaches 15 m/s from rest. Find Δp.
Solution path
Δp=0.2×15=3 N·s.
Momentum p=mv describes a body’s motion and has units kg·m/s. Impulse is the time integral of external force, with units N·s, equivalent to kg·m/s. It equals the momentum change Δp, not momentum itself. A brief large force can deliver the same impulse as a smaller force over longer time. State which bodies belong to the system, the time interval and the point about which angular momentum is taken. Momentum change depends on external impulse; internal interactions cancel in the total balance. Use SI units and distinguish momentum p from impulse J.
First compute momentum change from mass and velocity change. Divide by impact duration: Favg=Δp/Δt. A 0.20 kg ball going from rest to 15 m/s in 0.05 s receives Δp=3 N·s and experiences an average 60 N. Peak force may be higher. State which bodies belong to the system, the time interval and the point about which angular momentum is taken. Momentum change depends on external impulse; internal interactions cancel in the total balance. Use SI units and distinguish momentum p from impulse J.
Total momentum of a well-defined system is conserved when resultant external impulse over the interval is zero or negligible. Internal forces can accelerate individual bodies but do not change system momentum. For a collision, check whether external forces are small during the short contact time. State which bodies belong to the system, the time interval and the point about which angular momentum is taken. Momentum change depends on external impulse; internal interactions cancel in the total balance. Use SI units and distinguish momentum p from impulse J.
Angular momentum and torque are defined about a point; shifting the reference can change their values and equations. The relation ΔL=∫τdt uses the same point for L and τ. A useful choice is often a point where unknown forces exert no moment, but state that choice explicitly. State which bodies belong to the system, the time interval and the point about which angular momentum is taken. Momentum change depends on external impulse; internal interactions cancel in the total balance. Use SI units and distinguish momentum p from impulse J.
In a total-system balance, interaction forces occur in equal and opposite pairs and cancel when the system is chosen appropriately. For individual bodies, those same interactions are external and can change their momentum. A careful system boundary determines which forces enter and whether momentum conservation applies. State which bodies belong to the system, the time interval and the point about which angular momentum is taken. Momentum change depends on external impulse; internal interactions cancel in the total balance. Use SI units and distinguish momentum p from impulse J.
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Curated exam set for "Linear and angular impulse–momentum balance":
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Which formula describes Linear and angular impulse–momentum balance?
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How do you rearrange Linear and angular impulse–momentum balance for Average force?
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Which common mistake happens with Linear and angular impulse–momentum balance?
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