Forces and Newton's laws (Topic 2C)

A free-body diagram is a simplified representation of an object and the forces acting on it. It helps analyse the object’s dynamics by clearly showing all external forces acting on it, their directions and magnitudes.

Free-body diagrams are very useful for solving problems and understanding how different forces interact. Only the forces acting on the object are shown, not the forces the object exerts on other objects.

Forces are represented as arrows originating from the object, with:

• length indicating the relative magnitude of the force (longer arrow = greater force)
• direction indicating the direction in which the force acts.

Each force is labeled with its type and its magnitude if known.

Free body diagrams help in setting up equations using Newton’s laws of motion to solve for unknown forces or accelerations. They are crucial in problems involving objects on inclined planes, where forces must be resolved into components parallel and perpendicular to the incline.

When adding forces at an angle, always remember to label both the angle of application and the direction of the force with an arrow.

In the example above, the weight of a block on an inclined plane has been resolved into the weight force parallel to the slope, , and the weight perpendicular to the slope, .

Tension is the force exerted along a stretched object, such as a string, rope, or cable, when it is pulled tight by forces acting from opposite ends. It is uniform throughout an ideal massless, inextensible string. If the string has mass, the tension varies along its length.

Tension requires two forces acting in opposing directions on an object; this is what causes it to stretch and become taut.

An example of this is in a pulley system, where the tension in the rope transmits force between different objects, such as lifting a mass.

The normal contact force is the force exerted by a surface perpendicular to an object in contact with it. It arises from the contact between two solid surfaces and prevents objects from passing through each other. It is always directed perpendicular to the surface and acts away from it.

For example, a book resting on a table experiences a normal contact force equal to its weight. The table pushes up against the book, balancing the force of gravity.

The normal force is not always equal in magnitude to the weight of the object. It equals the weight only when the object is resting on a horizontal surface and no other vertical forces act on it. On an inclined plane, the normal force is less than the object’s weight.

Similarly, the normal force does not always act in the opposite direction to the weight. The normal force is always perpendicular to the surface.

Newton’s First Law of Motion is the law of inertia. It states that an object at rest or moving with constant velocity will remain so unless acted upon by a net external force.

The key idea is that objects resist change unless a net external force acts upon the object. Resisting changes in motion is called inertia.

The net force acting on an object is a single force that acts as the sum of all forces acting on an object. In this way, it describes the net effect of all the forces acting on an object. As each force acting on an object has a magnitude and direction, the net force is the vector addition of all of the forces present.

The direction and magnitude of the net force determine whether the object will accelerate:

• If the forces on an object are balanced (net force is zero), the object will not accelerate – it will either remain stationary or move at a constant velocity.

• If the forces are unbalanced (there is a nonzero net force), the object will accelerate in the direction of the net force.

If a constant force is applied to an object, the object will undergo a resulting acceleration, which will induce it to move. This motion can be studied in either one or two dimensions, such as along a flat surface or on an inclined plane.

• In one-dimensional motion, movement occurs either vertically (up and down) or horizontally (left and right).

• In two-dimensional motion, such as on a slope, both vertical and horizontal directions are involved. When dealing with slopes, it is often easier to resolve forces into parallel and perpendicular components rather than horizontal and vertical components.

A force acting at an angle to a surface can be split into two components.

Parallel component :

This is the component of the force acting along the surface (in the direction of motion on an incline).

Perpendicular component :

This is the component of the force acting perpendicular to the surface (typically balanced by the normal force in an inclined plane problem).

In the diagram above, the weight of a block on an inclined slope may be resolved into parallel and perpendicular components to the slope.

The normal force and act perpendicular to the slope, while acts parallel to the slope.

Newton’s Third Law of Motion is the law of action and reaction. It states that any action produces an equal and opposite reaction.

The key idea is that forces occur in pairs: the action of exerting a force produces an equal and opposite force. Equal and opposite are used to describe forces that have the same magnitude but are acting in opposite directions:

If only considering the magnitude of forces in a pair, and not direction, then:

An object in free fall is one that is falling solely under the influence of gravity. In the absence of air resistance, all objects experience the same acceleration due to gravity regardless of their mass.

David Scott famously proved this during the Apollo 15 mission to the moon, where a dropped hammer and feather reached the ground at the same time.

Mass vs. weight. In everyday language, someone might say, “I weigh ." However, this is technically incorrect:

• Mass is a scalar quantity measured in kilograms. It represents the quantity of matter that an object is made up of.
• Weight is a vector quantity. It is a force, measured in newtons, that an object experiences due to its location in a gravitational field.

An object’s mass is constant, but its weight varies depending on the strength of the gravitational field in which it is. For instance, the gravitational field strength on the Moon’s surface is , which means an object’s weight is approximately one-sixth what it would be on Earth.

The higher the mass and the smaller the radius of a celestial body, the greater the gravitational field strength at its surface.

Different celestial bodies have different values of , which refers to the gravitational field strength at the body’s surface. The gravitational field strength on a planet determines the force acting on an object or person due to gravity. The stronger the gravitational field strength, the heavier the object will feel, and the more force is needed to lift it.

In a gravitational field, the field lines:

• represent the direction and strength of a gravitational field,

• indicate the force on a mass at each point in a gravitational field,

• always point inward towards the mass, as gravity is always an attractive force, and

• never cross each other.

If the gravitational field lines are closer together, the field is stronger at that point. When the gravitational field lines are spaced further apart, the field is weaker. Gravitational field-line diagrams show that the field strength decreases with distance from the centre of mass.

A spherical mass produces a radial, symmetrical gravitational field, equivalent to a point mass at its centre. Non-spherical masses only approximate this pattern at large distances; close to the object, the field lines follow the actual shape of the mass distribution.

In a uniform gravitational field, the field lines are parallel and equidistant, indicating a constant gravitational field strength over the region.

The gravitational field close to the surface of a planet is approximately uniform. The gravitational field strength remains approximately constant over relatively small distances from the surface.

Although the planet’s gravitational field is radial, the distance between the test mass (the object experiencing gravity) and the centre of the planet does not change much relatively when you are near the surface of a planet.

An experiment to measure involving a steel ball-bearing, an electromagnet, a trap-door and a timer is shown below.

An electromagnet releases the steel ball, which triggers a timer to start. When the ball lands on the trapdoor, a trigger stops the timer and records the time taken.

The distance the ball falls can be measured with a ruler. The relevant equation of motion is:

We know that the ball starts from rest so and Therefore:

If the ball bearing is small, there will be some air resistance, but it should be small. The influence of air resistance can be minimised by ensuring the final velocity is not too high or by using a vacuum chamber.

Momentum is proportional to velocity. Velocity is a vector with a magnitude component (speed) and a direction. Hence, momentum is also a vector with both magnitude and direction.

Scalar quantities only have magnitude, such as mass and speed, and therefore do not change with direction.

The laws of classical mechanics dictate that momentum is always conserved. This means that the total momentum of all interacting bodies before and after a collision is the same.

The conservation of momentum can be used to calculate the velocity of objects before and after a collision. For instance, two objects colliding as illustrated below:

In order for the principle of conservation of momentum to be obeyed, the internal forces of a closed system must not change the total momentum. This is a consequence of Newton’s laws of motion.

Newton’s third law states that if object A exerts a force on object B, object B will exert an equal and opposite force on object A. For example, if you push against a wall, the wall will push back against you with a force of equal magnitude, so that it remains stationary:

Newton’s second law states that the net force is equal to the change in momentum over a time period

Therefore, if the net force equals zero, then the change in momentum equals zero, and momentum is conserved.

Imagine you’re using a wrench to tighten a bolt. To maximise the tightening effect, where on the wrench should you apply force?

The answer is at the end of the wrench. The reason behind this observation lies in what is called the moment of a force (torque).

By definition, the moment of a force (torque) is the turning effect of a force about some axis or pivot. The SI unit of the moment of a force is expressed in newton metres

An axis of rotation is simply an imaginary line about which a body rotates. In our example, the axis of rotation passes vertically through the centre of the bolt.

The moment of a force can be expressed as:

Where:

• is the magnitude of the force in newtons , and
• is the perpendicular distance from the line of action of the force to the axis of rotation (pivot) in metres .

The SI unit of the moment of a force is expressed in newton metres

The line of action of a force, as shown in the figure below, is an imaginary line that extends infinitely in both directions along the direction of the force vector.

When dealing with moments of force, we have three cases:

1. Perpendicular line of action
   In this case (diagram A), the line of action of the force is perpendicular to the axis of rotation. The moment of a force is easily calculated by multiplying the magnitude of the force by the line connecting the axis of rotation and the point of application of the force.
2. Non-perpendicular line of action
   Here, the force’s line of action is non-perpendicular (diagram B). Given the angle that the force makes with the horizontal, the perpendicular distance between the line of action and the axis of rotation should be calculated using trigonometric relations.
3. Line of action through the axis of rotation
   In the third case, the line of action of the force passes directly through the axis of rotation (diagram C). Consequently, the moment of the force is equal to zero, as there is no distance to create a turning effect.

It is useful to note that drawing all the forces acting on the object in question can help you see which forces are perpendicular to the distance from the pivot. Not all the forces will provide a turning effect, and it is not unusual for a question to provide more forces than required to mislead you.

The principle of moments states that for a body in equilibrium, the sum of clockwise moments about any point must equal the sum of counterclockwise moments about that same point.

For a body to remain in equilibrium, the total moment generated by all the forces causing clockwise rotation must balance the total moment generated by all the forces causing counterclockwise rotation:

Consider a rod that can rotate around a pivot , as illustrated in the figure above.

• Force produces a CW rotation.
• Force produces a CCW rotation.
• Force produces a CW rotation.
• Force produces a CCW rotation.

The rod is in equilibrium as the total moment around the pivot is balanced.

It is hard to study the motion of an irregular rigid body when a force is applied to it. In physics, to simplify calculations, we use a concept called the centre of mass (COM) to analyse the translational motion of an object. The centre of mass of an object is the point at which the mass of the object can be considered to be concentrated.

The centre of mass of an object:

• does not depend on the orientation of the body
• can be located outside or inside the physical boundaries of a body
• does not depend on the gravitational field strength.

The centre of mass of simple geometric shapes is located at their geometric centres and can be found through symmetry. The centres of mass of a circle, a box, a triangle, and a doughnut are represented below by a red dot.

To determine the centre of mass of any regular shape:

1. Start by dividing it into two equal parts using a dashed line.
2. Next, divide the same shape into equal parts using a different dashed line.
3. The intersection of the two dashed lines indicates the centre of mass.

The centre of gravity (COG) is the point through which an object’s entire weight is considered to act, the resultant of the small weight forces distributed throughout the object. Because moments depend on the location of a force’s line of action, this is the point used when calculating the turning effect of weight in equilibrium and rotational problems.

On a free-body diagram, an object’s weight is drawn as a single vector acting downward from its centre of gravity, rather than as multiple forces spread across the object.

The centre of gravity of an object has the following properties:

• It can be located outside or inside the physical boundaries of a body.
• It can change in a non-uniform gravitational field.

In cases where the gravitational field strength can be considered uniform, such as a small object near the surface of Earth, a body’s centre of mass coincides with its centre of gravity.

However, in cases involving a non-uniform gravitational field, such as a large celestial object in orbit of a massive body, the centre of mass (COM) is not in the same location as the centre of gravity (COG):

• Image A: the triangle’s COM is identical to its COG because the gravitational field acting on it is uniform.

• Image B: the Moon’s COM is at its geometric centre. However, the COG is closer to the Earth because the Moon’s gravitational field is significantly stronger on the side facing the Earth.

The location of the centre of mass of a body affects its stability. An object is:

• Stable if its centre of mass lies vertically above its base (image A).
• Unstable and will topple if its centre of mass lies vertically outside its base (image B).

Widening an object’s base or lowering its centre of mass increases its stability.

To determine the centre of gravity of an irregular-shaped object, complete the following steps:

1. Hang the body from a pivot near its edge and allow it to settle.
2. Use a market to draw a vertical line passing through the pivot.
3. Hang the body from a different pivot and allow it to settle.
4. Draw another vertical line passing through the new pivot.
5. The intersection of the two lines is the centre of gravity.

Since you are applying the procedure to a small object on the surface of the Earth, its centre of gravity is the same as its centre of mass.

Work done is defined as the energy transferred when a force moves an object through a distance:

Where:

• is the displacement in metres , and
• is the average force in the direction of the displacement in newtons .

Work done is measured in joules, One joule is the work done when a force of one newton moves an object one metre in the direction of the force.

For an object to move at a constant velocity, the net force acting on it must be zero. If an object has an opposing force, such as friction or gravity, it will require a continuous applied force to move at a constant velocity.

For an object moving at constant velocity, the work done by the applied force is equal to the work done against opposing forces like friction or gravity.

The direction of the force and the displacement of an object may not be the same. Generally, work done is given by:

Where:

• is displacement,
• is the average force, and
• is the angle between and

is the component of the force acting in the direction of motion.