Cambridge A-Level Mathematics 9709 - Mechanics (M2)

Forces: Motion of a body on a rough surface, connected particles, equilibrium of rigid bodies

This section covers the analysis of forces acting on objects, including motion on rough surfaces, systems of connected particles, and the conditions for equilibrium of rigid bodies. We will explore concepts such as friction, tension, and the application of Newton's laws to these scenarios.

1. Forces on a Rough Surface

When an object moves on a rough surface, it experiences several forces: gravity, the normal force, and the frictional force. The frictional force opposes the motion.

1.1 Static Friction

Static friction is the force that prevents an object from starting to move. It can vary in magnitude up to a maximum value.

The maximum static friction force ($f_{s,max}$) is given by:

$$f_{s,max} = \mu_s N$$ where:

• $\mu_s$ is the coefficient of static friction between the two surfaces.
• $N$ is the normal force between the surfaces.

An object will remain at rest or move with constant velocity as long as the applied force is less than or equal to the maximum static friction force.

1.2 Kinetic Friction

Kinetic friction is the force that opposes the motion of an object already in motion. It is generally less than static friction.

The kinetic friction force ($f_k$) is given by:

$$f_k = \mu_k N$$ where:

• $\mu_k$ is the coefficient of kinetic friction between the two surfaces.
• $N$ is the normal force between the surfaces.

1.3 Motion on a Horizontal Surface

Consider an object of mass $m$ on a horizontal surface with coefficient of static friction $\mu_s$ and coefficient of kinetic friction $\mu_k$. The normal force is equal to the weight of the object ($N = mg$).

If an external horizontal force $F$ is applied:

• If $F \le f_{s,max}$, the object will move with the applied force.
• If $F > f_{s,max}$, the object will move with kinetic friction.

2. Connected Particles

When particles are connected by strings or rods, the forces acting on each particle must be considered. We often use diagrams to represent these situations.

2.1 Example: Two Particles of Mass $m$ Connected by a String of Mass $m$

Consider two particles of mass $m$ connected by a light, inextensible string of mass $m$. The system is placed on a frictionless table and a horizontal force $F$ is applied to one of the particles.

The forces acting on the system are:

• The applied force $F$.
• The tension $T$ in the string.
• The weight of each particle $mg$.

Applying Newton's second law to the system:

$$m \frac{d^2x}{dt^2} = F - 2T$$ where $x$ is the displacement of the particles from their initial positions.

3. Equilibrium of Rigid Bodies

A rigid body is in equilibrium if the net force acting on it is zero and the net torque about any point is also zero.

3.1 Conditions for Equilibrium

For a rigid body to be in equilibrium, the following conditions must be met:

• The sum of all forces acting on the body must be zero: $$\sum \mathbf{F} = \mathbf{0}$$
• The sum of all torques about any chosen point must be zero: $$\sum \mathbf{τ} = \mathbf{0}$$

3.2 Example: A Rigid Body in Equilibrium

Consider a rigid body in equilibrium. We can choose any point on the body as the reference point for calculating torques. The torque $\tau$ is given by:

$$\tau = r \times F \sin \theta$$ where $r$ is the position vector from the reference point to the point where the force is applied, $F$ is the force vector, and $\theta$ is the angle between $r$ and $F$.

For equilibrium, the sum of all torques about the chosen point must be zero.

Topic	Key Concepts
Forces on a Rough Surface	Static friction, kinetic friction, coefficient of static friction ($\mu_s$), coefficient of kinetic friction ($\mu_k$), normal force ($N$), maximum static friction force ($f_{s,max}$).
Connected Particles	Tension, Newton's second law, system analysis.
Equilibrium of Rigid Bodies	Net force = 0, Net torque = 0, Torque calculation ($τ = r \times F \sin \theta$).