Calculation
ω′=(1+k)mlv/(J+ml²); v′=(ml²−kJ)v/(J+ml²)
A point mass strikes an initially stationary rigid body at distance l from its fixed pivot A. The impact can spin the body and slow or reverse the projectile.
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Calculate Impact on a Pivoted Rigid Body
A point mass strikes an initially stationary rigid body at distance l from its fixed pivot A. The impact can spin the body and slow or reverse the projectile.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Angular speed ω′ of pivoted body — Angular velocity immediately after impact in rad/s; positive in the direction caused by the incoming projectile. Use for subsequent motion or bearing-reaction analysis.
- Projectile velocity v′ — Signed projectile velocity immediately after impact in m/s. A negative value means rebound opposite to the incoming direction.
- Projectile mass m — Mass of the impacting object treated as a point mass, in kg, from weighing or a data sheet; strictly positive.
- Incoming speed v — Point-mass speed along the contact normal immediately before impact in m/s; positive toward the contact point.
- Mass moment of inertia J about pivot A — Moment of inertia of the complete struck body about the fixed pivot axis through A in kg·m², from CAD, geometry/mass or a test. Convert a centre-of-mass value with the parallel-axis theorem first.
- Impact lever arm l — Perpendicular distance from pivot A to the impact-impulse line in m. In this model it also equals the distance from A to the impact point along the body; positive.
- Coefficient of restitution k — Dimensionless ratio of relative separation to approach speed at the impact point, from a relevant impact test; 0 to 1.
Example
For m=1 kg, v=4 m/s, J=0.25 kg·m², l=0.5 m and k=0.5, denominator J+ml²=0.5 kg·m²; thus ω′=6 rad/s and v′=1 m/s. Angular momentum before and after impact is 2 kg·m²/s.
Assumptions and limits
Short frictionless single impact of a point mass along the contact normal on an initially stationary rigid body about a fixed frictionless pivot A. Contact-point normal speed is ω′l. Impact duration, pivot force, deformation, subsequent vibration and multiple contacts are excluded. Other geometries, moving bodies or tangential impulses require a different model.
Understand Impact on a pivoted rigid body
A small incoming object strikes a larger body away from its fixed pivot. The struck body starts rotating; the incoming object may continue or rebound.
What does this quantity describe?
The struck rigid body initially rests and can rotate only about fixed pivot A. Point mass m hits along the contact normal at distance l from A. The impact impulse has lever arm l. During the short impact, total angular momentum about A is conserved because the pivot force passes through A: mvl=mv′l+Jω′. Restitution coefficient k gives the relative separation speed at contact: ω′l−v′=kv. Solving yields ω′=(1+k)mlv/(J+ml²) and v′=(ml²−kJ)v/(J+ml²).
Formula and variables
ω′=(1+k)mlv/(J+ml²); v′=(ml²−kJ)v/(J+ml²)
Angular momentum about A: mvl=mv′l+Jω′Restitution at contact: ω′l−v′=kvAngular velocity: ω′=(1+k)mlv/(J+ml²)Projectile velocity: v′=(ml²−kJ)v/(J+ml²)
| Symbol / input | Meaning |
|---|---|
| Angular speed ω′ of pivoted body | Angular velocity immediately after impact in rad/s; positive in the direction caused by the incoming projectile. Use for subsequent motion or bearing-reaction analysis. |
| Projectile velocity v′ | Signed projectile velocity immediately after impact in m/s. A negative value means rebound opposite to the incoming direction. |
| Projectile mass m | Mass of the impacting object treated as a point mass, in kg, from weighing or a data sheet; strictly positive. |
| Incoming speed v | Point-mass speed along the contact normal immediately before impact in m/s; positive toward the contact point. |
| Mass moment of inertia J about pivot A | Moment of inertia of the complete struck body about the fixed pivot axis through A in kg·m², from CAD, geometry/mass or a test. Convert a centre-of-mass value with the parallel-axis theorem first. |
| Impact lever arm l | Perpendicular distance from pivot A to the impact-impulse line in m. In this model it also equals the distance from A to the impact point along the body; positive. |
| Coefficient of restitution k | Dimensionless ratio of relative separation to approach speed at the impact point, from a relevant impact test; 0 to 1. |
Choose the inputs correctly
m is positive projectile mass in kg (scale or data sheet). v is its positive speed along the contact normal immediately before impact in m/s (measurement or prior calculation). J is the mass moment of inertia of the struck body about fixed pivot A in kg·m² (CAD, geometry/mass or test), not automatically its centroidal value. l is the positive perpendicular distance from A to the impact line in m (drawing or measurement), here also the distance to the contact point. k is dimensionless restitution from a relevant test, between 0 and 1. ω′ is the struck body's angular velocity in rad/s; v′ is signed projectile velocity in m/s. Negative v′ means rebound.
How to use the calculator
Mark pivot A and the contact normal on a sketch. Confirm the normal lies at perpendicular distance l from A and the struck body starts at rest. Determine J about A, using the parallel-axis theorem if necessary. Enter m, v, l, J and k. Select angular velocity or projectile velocity in the result selector.
Worked example
With m=1 kg, v=4 m/s, J=0.25 kg·m², l=0.5 m and k=0.5, J+ml²=0.5 kg·m². Thus ω′=1.5·1·0.5·4/0.5=6 rad/s and v′=(0.25−0.125)·4/0.5=1 m/s. Check: angular momentum before is mvl=2 kg·m²/s, after is mv′l+Jω′=0.5+1.5=2 kg·m²/s; relative separation is ω′l−v′=2 m/s=k·v.
Understand the result and units
ω′ gives the struck body's angular motion immediately after impact. Positive v′ means continued motion; negative v′ means rebound. At k=0 the two contact points share the same velocity immediately after impact; this does not mean the entire struck body is stationary. At k=1 relative separation is highest in this ideal model.
Internally all quantities use kg, m, m/s, kg·m² and rad/s; k is dimensionless. Selected alternative units are converted to SI before calculation. Denominator J+ml² has units kg·m² and is positive for valid inputs.
Typical applications
Preliminary estimate for hinged levers, flaps or pendulum-mounted bodies hit normally by a small object. Pivot sizing and subsequent motion require further analysis.
Assumptions, limits and common mistakes
Only a point mass hitting an initially stationary rigid body about a fixed frictionless pivot, with contact normal perpendicular to the impact-point radius. No tangential impulse, friction, projectile spin, moving support or repeated contact. Pivot impact force may be large; linear momentum of both bodies is not conserved. k depends on contact and speed. No impact force or stress is calculated.
Common mistake: Use J about A, not the centre of mass. l must be the lever arm of the normal impulse; the radius length alone is insufficient for an oblique radius. Negative v′ is valid and means rebound. Conserve angular momentum about A, not linear momentum without including the pivot reaction.
Frequently asked questions
What is “Eccentric impact on a pivoted rigid body” used for?
Preliminary estimate for hinged levers, flaps or pendulum-mounted bodies hit normally by a small object. Pivot sizing and subsequent motion require further analysis.
Where do the input values come from?
m is positive projectile mass in kg (scale or data sheet). v is its positive speed along the contact normal immediately before impact in m/s (measurement or prior calculation). J is the mass moment of inertia of the struck body about fixed pivot A in kg·m² (CAD, geometry/mass or test), not automatically its centroidal value. l is the positive perpendicular distance from A to the impact line in m (drawing or measurement), here also the distance to the contact point. k is dimensionless restitution from a relevant test, between 0 and 1. ω′ is the struck body's angular velocity in rad/s; v′ is signed projectile velocity in m/s. Negative v′ means rebound.
What does the result not cover?
Only a point mass hitting an initially stationary rigid body about a fixed frictionless pivot, with contact normal perpendicular to the impact-point radius. No tangential impulse, friction, projectile spin, moving support or repeated contact. Pivot impact force may be large; linear momentum of both bodies is not conserved. k depends on contact and speed. No impact force or stress is calculated.