doPhysics · Centre of Mass and Collisions
Centre of Mass
The one point of a body or a system that moves like a single particle. Its position for point masses, rods, plates and solids, the standard results, composite bodies and bodies with holes, and its motion under external forces alone.
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Why this chapter
Why this chapter exists
Every earlier chapter treated a block, a car or a train as a single point, and wrote \(F = ma\) for it. A real body is made of countless particles, and when it spins or bends they all move differently. This chapter finds the one point that still obeys the particle laws: the centre of mass. It shows how to locate that point for any body, from two balls on a rod to a sphere with a cavity. Then it proves that only outside forces can move the point. That explains a man on a boat, two skaters on ice and a shell that bursts in flight.
You will learn
By the end of this chapter you will
- Locate the centre of mass of a set of point masses with the mass-weighted average, one coordinate at a time.
- Find the centre of mass of a rod, uniform or not, by integrating over elements \(dm\), and use symmetry to avoid the integral.
- Derive and use the standard results \(2R/\pi\), \(4R/3\pi\), \(R/2\), \(3R/8\) and \(h/4\).
- Find the centre of mass of a body built from pieces, and of a body with a hole or a cavity, by the negative-mass method.
- Show that the total momentum is \(M\vec v_{cm}\) and that \(M\vec a_{cm} = \vec F_{ext}\), because internal forces cancel in pairs.
- Use a fixed centre of mass to solve a man walking on a boat, skaters pulling on a rope and a shell bursting in flight.
Before you begin
Before you begin
From the chapters on Newton's laws you should know the second law, \(F = ma\), and the third law, with its equal and opposite pairs. From the chapter on Accelerated Objects you should know momentum, \(p = mv\). From Motion in Two Dimensions you should know the parabola of a projectile. You should be able to integrate powers of \(x\) and \(\sin\theta\).
Lessons
1
The Point That Moves Like a Particle
A tumbling body hides one point that follows a clean parabola. That point is the centre of mass, the mass-weighted average of the positions. Two masses on a line, then many masses in a plane, one coordinate at a time.
One point follows a parabolaCentre of mass of two particlesCentre of mass of many particles
2
Centre of Mass of Continuous Bodies
A rod or a plate is a continuous spread of mass. Cut it into small elements and the sums become integrals. The uniform rod, a rod whose density changes along it, and the symmetry that makes most integrals unnecessary.
From sum to integralRod of varying densitySymmetry fixes the centre of mass
3
Standard Shapes and Composite Bodies
Five results every student must know, each derived once. They are the semicircular ring \(2R/\pi\), the semicircular disc \(4R/3\pi\), the hemispherical shell \(R/2\), the solid hemisphere \(3R/8\) and the solid cone \(h/4\). Then bodies built from pieces, each piece replaced by a point at its own centre of mass.
Semicircular ring and semicircular discHemispheres and the solid coneComposite bodies
4
Bodies with Holes
A plate with a hole is a full plate plus a negative mass where the hole is. A hole at the rim of a disc, a cavity in a ball, a cut corner, and a weight that balances a wheel.
The negative mass methodCavities in solidsCorners cut and weights added
5
Motion of the Centre of Mass
Differentiate the definition once and the total momentum appears as \(M\) times the velocity of the centre of mass. Differentiate again and the internal forces cancel in pairs, leaving \(Ma_{cm} = F_{\text{ext}}\). Two blocks on a spring and the Atwood machine, each seen as one body.
Velocity of the centre of massInternal forces cancel in pairsThe system as one body
6
When No External Force Acts
With no net external force the centre of mass keeps its velocity, and a centre of mass at rest stays at rest. A man walking on a boat, two skaters pulling on a rope, and a shell that bursts at the top of its flight.
The centre of mass keeps its velocityPulling pushing and walkingThe exploding shell
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- axis of symmetry
- A line about which a body is a mirror image of itself. For a uniform body the centre of mass lies on every axis of symmetry.
- cavity
- A hollow space inside a solid body. For a uniform solid, the mass removed by a cavity is proportional to its volume.
- centre of mass
- The point whose position is the mass-weighted average of the positions of all the parts of a body or system, \(\vec r_{cm} = \sum m_i\vec r_i / M\). It moves like a single particle of the total mass under the net external force.
- centroid
- The geometric centre of a shape, the plain average of the positions of its points. For a triangle it is where the three medians meet. For a uniform body it coincides with the centre of mass.
- composite body
- A body made of simple pieces, such as rods, plates and hemispheres. Its centre of mass is found by replacing each piece by its mass placed at its own centre of mass.
- element of mass
- A piece of a continuous body so small that it can be treated as a point, of mass \(dm\). Sums over particles become integrals over such elements.
- external force
- A force on part of a system from a body outside it, such as gravity, an outside push or the friction of the ground. Only external forces can accelerate the centre of mass.
- internal force
- A force exerted by one part of a system on another part of the same system. Internal forces come in equal and opposite pairs and cannot move the centre of mass.
- isolated system
- A system on which the net external force is zero. The velocity of its centre of mass stays constant.
- lamina
- A thin flat plate whose thickness can be ignored, such as a sheet of metal or cardboard. Its mass is described by the mass per unit area.
- linear mass density
- The mass per unit length of a thin rod or wire, \(\lambda = dm/dx\), in kg/m. An element of length \(dx\) has mass \(dm = \lambda\,dx\).
- negative mass
- A bookkeeping device for a body with a piece removed. The removed piece enters the particle formula with a minus sign, at its own centre of mass. No real mass is negative.
- surface mass density
- The mass per unit area of a lamina, \(\sigma = dm/dA\), in kg/m². An element of area \(dA\) has mass \(dm = \sigma\,dA\).
- total momentum
- The vector sum of the momenta of all the particles of a system, \(\vec P = \sum m_i\vec v_i\). It equals the total mass times the velocity of the centre of mass, \(M\vec v_{cm}\).
- uniform
- Having the same density at every point. A uniform rod has the same mass per unit length along its whole length, so \(\lambda = M/L\).