Physics Study Notes: Forces, Newton's Laws, and Momentum

Forces and Newton's Laws of Motion

Understanding Forces and Tension

Forces are vector quantities that cause objects to accelerate according to Newton's Second Law. Tension is a specific force transmitted through a string, rope, or wire when it is pulled tight by forces acting from opposite ends.

• Newton's Second Law: The net force on an object is equal to the mass of the object multiplied by its acceleration.

$F_{\text{net}} = m a$

• Tension in a Wire: When a mass hangs from a wire, the tension equals the weight of the mass if the system is in equilibrium.

• Horizontal and Vertical Components: Forces can be resolved into horizontal (x) and vertical (y) components using trigonometry.

Example: A 35 kg mass is suspended by a wire and a guy wire at an angle. To find the tension and force components, use vector resolution and equilibrium conditions.

Forces on an Inclined Plane

When an object is on an inclined plane, gravity acts downward, and the normal force acts perpendicular to the surface. Additional forces, such as applied forces or friction, may also be present.

• Resolving Forces: The weight can be split into components parallel and perpendicular to the incline.

$F_{\parallel} = m g \sin \theta$ $F_{\perp} = m g \cos \theta$

Example: A diver of mass 75 kg stands at the right end of a board supported by two forces. To find the magnitude of the forces, apply equilibrium conditions (sum of forces and torques equals zero).

Spring Force and Hooke's Law

Springs exert a restoring force proportional to their displacement from equilibrium, described by Hooke's Law.

• Hooke's Law: The force exerted by a spring is proportional to its extension or compression.

$F = -k x$

• Spring Constant (k): A measure of the stiffness of the spring (units: N/m).

• Weight Supported by a Spring: At equilibrium, the spring force balances the weight of the object.

Example: A spring scale is used to measure weight. If a spring with constant 1 N/m stretches 8.5 cm, the weight is $F = kx = 1 \times 0.085 = 0.085$ N.

Momentum and Impulse

Impulse and Change in Momentum

Impulse is the product of force and the time over which it acts. It equals the change in momentum of an object.

• Impulse-Momentum Theorem: The impulse on an object equals its change in momentum.

$J = F \Delta t = \Delta p = m v_f - m v_i$

• Application: In sports, the impulse delivered to a ball changes its velocity.

Example: A ball of mass 0.145 kg is hit by a bat, changing its speed from 0 to 45.0 m/s. The impulse is $J = m \Delta v = 0.145 \times 45.0 = 6.525$ kg·m/s.

Collisions and Conservation of Momentum

In the absence of external forces, the total momentum of a system remains constant during a collision.

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not.

$\sum m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

Example: Two skaters collide and hold on to each other. Their final velocity can be found using conservation of momentum.

Applications: Vehicles and Projectiles

Momentum principles apply to vehicles, projectiles, and objects in motion.

• Stopping Distance: The distance required for a vehicle to stop depends on its initial speed and the frictional force.

• Projectile Motion: The final speed of a ball hit off a tee can be found using impulse and momentum change.

Example: A golf ball of mass 0.046 kg is hit with an impulse of 0.065 N·s. The final speed is $v = \frac{J}{m} = \frac{0.065}{0.046} \approx 1.41$ m/s.

Summary Table: Key Quantities and Formulas

Quantity	Symbol	Formula	Units
Force	F	$F = m a$	Newtons (N)
Weight	W	$W = m g$	Newtons (N)
Spring Force	F	$F = -k x$	Newtons (N)
Impulse	J	$J = F \Delta t$	Newton-seconds (N·s)
Momentum	p	$p = m v$	kg·m/s