Forces on an Inclined Plane

Analyzing Contact Forces on an Inclined Ramp

When an object of mass M rests on an inclined plane at angle θ, it experiences several forces. Understanding these forces is essential for solving problems involving equilibrium and friction.

• Normal Force (FN): The force perpendicular to the surface, balancing the component of gravity normal to the plane.

• Friction Force (Ff): The force parallel to the surface, opposing motion or potential motion.

• Gravitational Force (Fg): Acts vertically downward, with components both parallel and perpendicular to the plane.

Equilibrium Conditions:

• Sum of forces in the x-direction (parallel to the plane):

• Sum of forces in the y-direction (perpendicular to the plane):

• At rest: ,

Solving for Forces:

• Normal force:

• Friction force (if at rest):

• If friction is limiting:

Direction and Magnitude:

• The contact force from the ramp is the vector sum of normal and friction forces.

• Magnitude:

• Direction: , where is the angle from the normal.

Example: For , kg, m/s2:

• N

• N

Forces on a Sled: Friction and Pulling Force

Pulling a Sled at Constant Velocity

When pulling a sled across a horizontal surface at constant velocity, the applied force must balance the kinetic friction force.

• Applied Force (Fpx): The horizontal force exerted by the rope.

• Kinetic Friction Force (Fkf):

• Normal Force (FN): (if the rope is horizontal)

Equilibrium Conditions:

• Sum of forces in x-direction:

• At constant velocity:

Solving for Applied Force:

Example: For kg, , m/s2:

• N

Drag Forces and Terminal Velocity

Pressure Drag on Large Objects

For large objects moving through non-viscous fluids, pressure drag is the dominant resistive force. It acts opposite to the direction of motion.

• Pressure Drag Force:

• Cd: Drag coefficient (dimensionless)

• ρ: Fluid density

• A: Cross-sectional area

• v: Velocity of the object

Terminal Velocity of a Skydiver

When a skydiver falls, they accelerate until the drag force equals their weight, reaching terminal velocity (vt).

• Free Body Diagram: Forces are gravity (down) and drag (up).

• Equilibrium at Terminal Velocity:

• Solving for Terminal Velocity:

Example: For kg, , kg/m3, m2:

• m/s

Skydiver Descent and Parachute Deployment

After reaching terminal velocity, a skydiver opens their parachute, increasing the cross-sectional area and drag coefficient, which rapidly decreases their velocity.

• New Drag Force: If drag increases by a factor of 5,

• Net Force at Deployment:

Example: If , then , so (upward acceleration).

Summary Table: Forces in Different Scenarios

Additional info: The FANCL checklist referenced in the notes stands for Free body diagram, Axes, Newton's laws, Components, and Links to equations. This systematic approach helps organize force analysis in physics problems.