🔋College Physics I – Introduction Unit 9 – Statics and Torque

Key Concepts and Definitions

• Statics studies forces acting on objects at rest or in equilibrium
• Force is a push or pull on an object measured in Newtons (N)
• Equilibrium occurs when the net force and net torque acting on an object are zero
  • Static equilibrium objects are at rest
  • Dynamic equilibrium objects move at constant velocity
• Torque is the rotational equivalent of force causing an object to rotate measured in Newton-meters (N⋅m)
• Center of mass is the average position of all the mass in a system
• Center of gravity is the point where the force of gravity appears to act on an object
• Moment arm is the perpendicular distance from the line of action of a force to the axis of rotation
• Couple is a pair of equal and opposite forces that cause pure rotation without translation

Forces and Equilibrium

• Forces can be represented as vectors with magnitude and direction
• Free-body diagrams illustrate all the forces acting on an object
• Newton's first law states that an object at rest stays at rest and an object in motion stays in motion with constant velocity unless acted upon by an unbalanced force
• For an object to be in equilibrium, the net force in all directions must be zero (ΣFx = 0, ΣFy = 0, ΣFz = 0)
  • Translational equilibrium no net force
  • Rotational equilibrium no net torque
• Normal force is the force exerted by a surface on an object perpendicular to the surface
• Tension is the force exerted by a rope, cable, or string pulling on an object
• Friction is the force resisting the motion of an object sliding along a surface

Torque and Rotational Motion

• Torque causes an object to rotate about an axis
  • Clockwise torques are considered negative
  • Counterclockwise torques are considered positive
• The magnitude of torque depends on the force applied and the moment arm: $\tau = F \times r$
• Net torque is the sum of all torques acting on an object: $\tau_{net} = \tau_1 + \tau_2 + ... + \tau_n$
• For an object to be in rotational equilibrium, the net torque must be zero: $\tau_{net} = 0$
• Torque is a vector quantity with both magnitude and direction
• The right-hand rule determines the direction of torque
  • Point your fingers in the direction of the force
  • Curl your fingers towards the axis of rotation
  • Your thumb points in the direction of the torque vector

Center of Mass and Gravity

• Center of mass is the point representing the average position of all the mass in a system
  • For symmetrical objects, the center of mass is at the geometric center (a uniform rod)
• Center of gravity is the point where gravity appears to act on an object
  • For objects near Earth's surface, the center of gravity is approximately the same as the center of mass
• The center of mass can be calculated using the formula: $\vec{r}_{cm} = \frac{\sum_{i} m_i \vec{r}_i}{\sum_{i} m_i}$
• An object's weight acts through its center of gravity
• Stable equilibrium occurs when an object's center of gravity is directly above its base of support
• Unstable equilibrium occurs when an object's center of gravity is directly above its pivot point

Solving Statics Problems

• Identify the object or system in equilibrium
• Draw a free-body diagram showing all the forces acting on the object
  • Choose a convenient coordinate system (usually x and y axes)
  • Label all forces with their magnitudes and directions
• Write equations for the net force and net torque in each direction
  • ΣFx = 0, ΣFy = 0, ΣFz = 0 for translational equilibrium
  • Στ = 0 for rotational equilibrium
• Solve the equations for the unknown quantities (forces, angles, or distances)
  • Use trigonometry and geometry to relate angles and distances
  • Substitute known values and simplify the equations
• Check your answers for reasonableness and consistency with the problem statement

Real-World Applications

• Bridges and buildings must be designed to support their own weight and any additional loads (vehicles, people, wind)
• Cranes use torque to lift heavy objects by applying a smaller force over a larger moment arm
• Levers and pulleys use the principle of torque to multiply force and change its direction
  • A see-saw is an example of a lever
  • A block and tackle is an example of a pulley system
• Biomechanics applies statics to the human body (analyzing forces on joints and muscles)
• Vehicles (cars, airplanes, ships) must be designed with their center of gravity in mind for stability and control
• Balancing objects (a person walking on a tightrope, a waiter carrying a tray) requires an understanding of center of mass and equilibrium

Common Mistakes and Tips

• Not drawing a complete and accurate free-body diagram
  • Include all forces acting on the object
  • Use the correct magnitudes and directions for each force
• Confusing the signs of torques (clockwise is negative, counterclockwise is positive)
• Forgetting to set the net force and net torque equal to zero for equilibrium
• Using the wrong moment arm (the perpendicular distance from the force to the axis of rotation)
• Not double-checking units and significant figures in your answers
• Practice solving problems step-by-step to develop a consistent approach
• Analyze the problem statement carefully to identify the key information and unknowns
• Use sketches and diagrams to visualize the problem and organize your solution

Practice Problems and Examples

• A uniform beam of length 6 m and weight 200 N is supported by a pin at one end and a cable at the other end. The cable makes an angle of 30° with the horizontal. Find the tension in the cable and the force exerted by the pin.
• Two people are carrying a 50 kg crate by holding opposite ends of a 2 m long bar. If the crate is 0.5 m from one end of the bar, how much weight does each person support?
• A 10 m long ladder weighing 200 N leans against a frictionless wall at an angle of 60° to the horizontal. The base of the ladder is 4 m from the wall. Find the normal force exerted by the ground on the ladder and the force exerted by the wall on the ladder.
• A 2 kg block sits on a 30° incline. The coefficient of static friction between the block and the incline is 0.4. Find the minimum and maximum forces that can be applied parallel to the incline without causing the block to move.
• A uniform 4 m long plank weighing 300 N is supported by two vertical ropes attached 1 m from each end. A 50 kg person stands 1.5 m from the left end of the plank. Find the tension in each rope.