3.5 Centres of Mass

• What is the definition of the centre of mass (CoM) of an object or system of particles?

  Average position of mass
• What is the key difference between the CoM and the CoG?

  CoM depends on mass, CoG on gravity
• Match the feature with the correct concept:

  Definition of CoM ↔️ Average position of mass

  Dependence of CoM location ↔️ Mass distribution

  Use of CoM ↔️ Balance, stability, dynamics
• For what purpose is the CoM used in mechanics?

  Balance, stability, dynamics
• The location of the CoM depends on the gravity field.

  False
• Arrange the properties of the CoM in a logical order:

  1️⃣ No direct force acts at the CoM

  2️⃣ CoM location depends on mass distribution

  3️⃣ CoM is used for balance, stability, and dynamics
• The location of the CoG depends on the gravity field
• \(\mathbf{r}_{CoM}\) represents the position vector of the centre of mass.


  True
• The CoM calculation for particles considers both mass and position.


  True
• Match the property with its context:

  Sum of mass-weighted positions ↔️ System of particles

  Integral of position multiplied by density ↔️ Continuous distribution
• What is the centre of mass (CoM) of an object?

  The point of perfect balance
• What are the uses of the CoM for an object?

  Balance, stability, dynamics
• What happens to an object's stability if the CoM moves outside the base of support?

  It becomes unstable
• What does \(m_i\) represent in the formula for calculating the CoM of a system of particles?

  Mass of the i-th particle
• To calculate the CoM for a continuous distribution, the density function \rho(\mathbf{r}) is used.
• Unlike discrete particles, finding the CoM of a continuous distribution requires integration.


  True
• The CoM of a uniform rod of length \(L\) is located at \(L/2\).

  True
• If the CoM is directly above the base of support, the object is stable
• Why does no direct force act at the CoM?

  It's the average position of mass
• The stability of an object increases when its CoM is outside the base of support.

  False
• What type of force acts directly at the centre of gravity (CoG)?

  Gravity
• What is the definition of the centre of mass (CoM)?

  Average position of mass
• What is the primary use of the CoM?

  Balance, stability, dynamics
• What is the definition of the centre of gravity (CoG)?

  Average position of weight
• In the CoM formula, \(n\) represents the total number of particles
• What is the primary difference in calculating the CoM for continuous distributions versus systems of particles?

  Continuous distributions use integration
• What determines the location of the CoM in an object?

  Distribution of mass
• The centre of gravity (CoG) is the average position of an object's weight.

  True
• The CoM location depends on the mass distribution within the object.


  True
• The CoM of a system of particles is the point where the system would balance if suspended.

  True
• What is the CoM of a system with total mass \(10\) kg and a total mass x position vector of \(4\mathbf{i} + 13\mathbf{j}\)?

  0.4\mathbf{i} + 1.3\mathbf{j}
• Match the feature with the correct concept:

  Average position of mass ↔️ Centre of Mass

  Gravity acts at this point ↔️ Centre of Gravity

  Dependent on mass distribution ↔️ Location of CoM
• The density function in the CoM integral is denoted as \rho(\mathbf{r})
• The formula for the CoM of a system of particles is \mathbf{r}_{CoM} = \frac{\sum m_i \mathbf{r}_i}{\sum m_i}
• Steps to calculate the CoM of a composite body

  1️⃣ Identify and separate each component

  2️⃣ Determine the mass and CoM of each component

  3️⃣ Sum the products of each component's mass and CoM position

  4️⃣ Divide by the total mass of the composite body
• A car is an example of a composite body.
• What happens to an object's stability if its CoM moves outside its base of support?

  Becomes unstable
• Match the application with its example:

  Design and Engineering ↔️ Vehicle stability

  Balance and Stability ↔️ Pencil standing upright

  Dynamics ↔️ Projectile motion
• What is a composite body formed by?

  Combining individual shapes
• The composition of a composite body is always homogeneous.

  False