Gravitational Force, Weight, and Normal Force: Study Notes

Gravitational Force and Weight

Definition and Nature of Gravitational Force

The gravitational force is a fundamental force of attraction that exists between all pairs of masses. In most introductory physics problems, one of the masses is the Earth, resulting in a downward force experienced by all objects near its surface.

• Force of Attraction: Gravitational force acts to pull masses toward each other.

• Direction: Near Earth's surface, this force is directed downward, toward the center of the Earth.

• Free Fall: Motion under the influence of only gravitational force is called free fall. All objects in free fall experience the same acceleration, regardless of their mass.

Weight vs. Mass

Weight is the force exerted on a mass by gravity. It is not the same as mass, although the terms are often used interchangeably in everyday language.

• Definition: The weight of an object of mass m is given by:

$w = m \vec{g}$

• Mass: A measure of the amount of matter in an object (scalar quantity, measured in kilograms).

• Weight: The gravitational force acting on the object (vector quantity, measured in newtons).

• Key Point: Weight depends on the local value of gravitational acceleration g, which varies with location (e.g., Earth, Mars, or in orbit).

Gravitational Acceleration and Field Strength

The acceleration due to gravity near Earth's surface is approximately $9.8\,\text{m/s}^2$. This value is derived from the universal law of gravitation:

$g = \frac{GM_E}{r_E^2}$

• $G$: Universal gravitational constant ($6.67 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2$)

• $M_E$: Mass of Earth ($5.98 \times 10^{24}\,\text{kg}$)

• $r_E$: Radius of Earth ($6400\,\text{km}$)

Gravitational field strength is defined as the gravitational force per unit mass:

$g = \frac{F}{m}$

• Direction of $\vec{g}$: Always points toward the center of the Earth (downward near the surface).

• Direction of $\vec{w}$: Same as $\vec{g}$, downward.

Weight on Different Celestial Bodies

The value of g and thus the weight of an object depends on the mass and radius of the celestial body.

Body	Mass (kg)	Radius (km)	g (m/s2)	Weight of 100 kg Astronaut (N)
Earth	5.98 × 1024	6400	9.8	980
Mars	6.42 × 1023	3400	3.71	371

Example: On Earth, a 100 kg astronaut has a weight of $w = mg = 980\,\text{N}$. On Mars, the same astronaut's weight is $w = mg = 371\,\text{N}$.

Weight in Low Earth Orbit

In low Earth orbit (about 300 km above the surface), the gravitational acceleration is slightly less than on the surface:

$w = m g(r_o) = m \left( \frac{GM_E}{(R_E + h)^2} \right)$

• Example: For a 100 kg astronaut in low Earth orbit, $w \approx 890\,\text{N}$, about 10% less than on the surface.

• Key Point: Astronauts in orbit are not truly weightless; their weight is reduced but not zero.

Normal Force: A Contact Force

Definition and Properties

The normal force is a contact force that acts perpendicular to the surface of contact between two objects. It arises from the electromagnetic interactions between atoms in the surfaces.

• Direction: Always perpendicular to the contact surface.

• Origin: Results from the repulsion of atoms when objects are pressed together.

Normal Force on Flat and Inclined Surfaces

On a flat surface, the normal force balances the weight of the object:

$\sum F_y = N - w = 0 \implies N = w = mg$

• Key Point: The normal force equals the magnitude of the weight only when the surface is horizontal and there are no other vertical forces.

• Not a Newton's Third Law Pair: The normal force and weight act on the same object and are not action-reaction pairs.

Normal Force on an Inclined Plane

When an object is on an inclined plane, the weight can be resolved into components perpendicular and parallel to the surface:

• Component perpendicular to the surface: $mg \cos \theta$

• Component parallel to the surface: $mg \sin \theta$

Normal force: $N = mg \cos \theta$

Example: For a block on a roof inclined at $30^\circ$:

• Normal force: $N = mg \cos 30^\circ$

• Downward force along the roof: $mg \sin 30^\circ$

• If no other forces oppose gravity, the block accelerates down the roof with $a = g \sin 30^\circ$

Summary Table: Weight and Normal Force on Different Surfaces

Situation	Normal Force ($N$)	Weight ($w$)	Acceleration
Flat surface	$mg$	$mg$	0
Inclined plane ($\theta$)	$mg \cos \theta$	$mg$	$g \sin \theta$ (down the incline)

Additional info:

• In all cases, the direction of the weight is vertically downward, while the normal force is perpendicular to the contact surface.

• On an inclined plane, the normal force is less than the weight due to the angle of the surface.

• Weight varies with location due to changes in gravitational acceleration.