Forces, Friction, and Newton's Laws: Physics Exam Study Notes

Forces and Newton's Laws

Weight and Gravitational Acceleration

The weight of an object is the force exerted on it by gravity. This force depends on the object's mass and the local acceleration due to gravity.

• Definition: Weight ($W$) is given by $W = mg$, where $m$ is mass and $g$ is gravitational acceleration.

• Example: On the Moon, $g = 1.62\,\text{m/s}^2$. A 99 kg astronaut weighs $W = 99 \times 1.62 = 160.38\,\text{N}$.

Force and Acceleration on Frictionless Surfaces

When a force is applied to an object on a frictionless surface, Newton's Second Law ($F = ma$) determines its acceleration.

• Formula: $a = \frac{F}{m}$

• Example: A 10 kg block with a 53.5 N force: $a = \frac{53.5}{10} = 5.35\,\text{m/s}^2$

Frictional Forces

Friction opposes the motion of objects in contact. The frictional force ($f$) is calculated as $f = \mu N$, where $\mu$ is the coefficient of friction and $N$ is the normal force.

• Static friction: Prevents motion until a threshold force is exceeded.

• Kinetic friction: Acts when objects are sliding.

• Example: A 75 N object pulled with a 50 N horizontal force, friction force is 30 N. Net force: $50 - 30 = 20\,\text{N}$, so $a = \frac{20}{7.5} = 2.67\,\text{m/s}^2$

Applications of Newton's Laws

Crate Pulled by a Force at an Angle

When a force is applied at an angle, only the horizontal component contributes to acceleration.

• Horizontal component: $F_x = F \cos \theta$

• Example: 100 N at 20°: $F_x = 100 \cos 20° = 93.97\,\text{N}$; $a = \frac{93.97}{30} = 3.13\,\text{m/s}^2$

Equilibrium and Net Force

When an object moves at constant velocity, the net force is zero. The sum of all forces in each direction must balance.

• Example: A block pulled by three forces: $F_1 = 38\,\text{N}$, $F_2 = 26\,\text{N}$, $F_3 = 52\,\text{N}$, with $F_3$ at 260° from the x-axis.

Picture Hanging by Two Wires

When an object is suspended by two wires at different angles, the tension in each wire can be found using equilibrium conditions.

• Example: 10.0 kg picture, wires at 52° and 40°: Tension in the first wire is $T_1 = 98.0\,\text{N}$.

Pulley Systems and Tension

Force Required to Lift an Object

To lift an object at constant speed, the applied force must equal the object's weight.

• Formula: $F = mg$

• Example: Piano of 200 kg: $F = 200 \times 9.8 = 1960\,\text{N}$ (but answer given is 1900 N, likely rounded).

Multi-Block Pulley Systems

In systems with multiple blocks and pulleys, analyze forces and accelerations using Newton's laws and constraints from the ropes.

• Example: Block A pulled down with $F = 60\,\text{N}$, acceleration $a = 2.5\,\text{m/s}^2$; tension in rope is $T = 190\,\text{N}$, mass of block C is $4.9\,\text{kg}$.

Friction on Inclined Planes

Static and Kinetic Friction on Ramps

When objects are on inclined planes, friction and gravity both affect motion. The minimum force to move an object up a ramp depends on the angle and coefficient of friction.

• Static friction: $f_s = \mu_s N$

• Normal force on ramp: $N = mg \cos \theta$

• Example: 5.0 kg box on 37° ramp, $\mu = 0.40$; acceleration $a = 2.0\,\text{m/s}^2$ down the ramp.

Minimum Force to Move an Object Up a Ramp

To start moving a package up a ramp, the applied force must overcome both gravity and static friction.

• Formula: $F_{min} = mg \sin \theta + \mu_s mg \cos \theta$

• Example: 5.0 kg package, $\theta = 30°$, $\mu_s = 0.40$; $F_{min} = 29\,\text{N}$

Motion on Curved Paths

Circular Motion and Friction

For vehicles on curved roads, the maximum safe speed depends on the radius of the curve and the coefficient of static friction.

• Formula: $v_{max} = \sqrt{\mu_s g r}$

• Example: $\mu_s = 0.60$, $r = 790\,\text{m}$; $v_{max} = 68\,\text{km/h}$

Summary Table: Key Formulas and Concepts

Concept	Formula	Example
Weight	$W = mg$	Moon: $W = 99 \times 1.62 = 160.38\,\text{N}$
Newton's Second Law	$F = ma$	10 kg block, $F = 53.5\,\text{N}$: $a = 5.35\,\text{m/s}^2$
Frictional Force	$f = \mu N$	75 N object, $\mu$ inferred from $f = 30\,\text{N}$
Inclined Plane Normal Force	$N = mg \cos \theta$	5 kg box, $\theta = 37°$
Minimum Force Up Ramp	$F_{min} = mg \sin \theta + \mu_s mg \cos \theta$	5 kg, $\theta = 30°$, $\mu_s = 0.40$
Max Speed on Curve	$v_{max} = \sqrt{\mu_s g r}$	$\mu_s = 0.60$, $r = 790\,\text{m}$

Additional info: Some values and context were inferred from standard physics principles and the provided answers. These notes cover core concepts in Newtonian mechanics, including forces, friction, equilibrium, and circular motion, as relevant for introductory college physics.