Forces and Equilibrium Notes

Normal Force

• Definition: Electrostatic force exerted by a surface on an object, balancing the force exerted by the object on the surface.
• Direction: Always acts perpendicular and upwards from the surface on which the object is situated.
• Balance: Balances the component of weight directed along the same axis but in the opposite direction.
• Horizontal Ground, Object at Rest: $F_N = mg$.
• Stationary Object with Other Vertical Forces: Find $F_N$ by equating all forces acting perpendicular to the ground.
• Object on Incline, Stationary: $F_N$ balances the component of $mg$ parallel to the normal force.
• Object on Incline, Stationary, with Other Perpendicular Forces: $F_N$ balances the component of $mg$ parallel to the normal force plus/minus other forces acting perpendicular to the incline.

Force of Tension ($F_T$)

• Definition: Force exerted by a rope on an object.
• Direction: Always in the direction in which the rope is pulled.

Force of Friction ($F<em>{fs}$ and $F</em>{fk}$)

• Cause: Arises from the unevenness in the surfaces of objects, even highly-polished surfaces. Atoms stacked side by side create bumps, which lead to friction.
• Types:
  • Static friction ($F_{fs}$): For a body not in motion.
  • Kinetic friction ($F_{fk}$): For a body in motion.
• Static Friction in Response to Applied Force:
  • Static friction acts in response to an applied force ($F_T$).
  • As $F<em>T$ increases, $F</em>{fs}$ also increases until it reaches its maximum possible value ($F_{fs,max}$).
  • If $F<em>T$ gets bigger than $F</em>{fs,max}$, the object starts to move.
• Computation of Static Friction: When asked to compute $F<em>{fs}$, compute $F</em>{fs,max}$.

Kinetic and Static Friction Equations

• Kinetic Friction: Always opposite to the direction of motion.
  • $F<em>{fk} = u</em>k F_N$
• Static Friction:
  • $F<em>{fs,max} = u</em>s F_N$
• $<br /> u<em>s$ and $u</em>k$: Coefficients of static and kinetic frictions, respectively; depend on the two surfaces in contact.

Equilibrium System

• Definition: A system is in equilibrium when the sum of all forces on the system is zero ($F_{net} = 0$).
• Condition: No acceleration.
• State: Either not moving or moving with a constant velocity.
• Types:
  • Static equilibrium: Object is at rest.
  • Dynamic equilibrium: Object moving in a straight line at a constant speed (i.e., constant velocity).

Equilibrium Problem Solving Strategy

1. PREPARE
   • Make simplifying assumptions.
   • Check that the object is either at rest or moving with constant velocity ($a = 0$).
   • Identify forces and show them on a free-body diagram.
2. SOLVE
   • Use Newton's Second Law in component form:
     • $\Sigma F<em>x = ma</em>x = 0$
     • $\Sigma F<em>y = ma</em>y = 0$
   • Read the components from the free-body diagram.
3. ASSESS
   • Is the result reasonable?

Non-Equilibrium or Dynamic Systems

• Condition: Net force is not zero ($F_{net} = ma$).
• State: Possesses a non-zero acceleration (speeding up or slowing down).
• Note: Dynamic equilibrium system is not the same as dynamic systems.

Dynamic Problem Solving Strategy

1. PREPARE
   • Make simplifying assumptions.
   • Make a visual overview:
     • Sketch a pictorial representation.
     • Identify known quantities and what the problem is trying to find.
   • Identify all forces and show them on a free-body diagram.
2. SOLVE
   • Use Newton's Second Law in component form:
     • $\Sigma F<em>x = ma</em>x$
     • $\Sigma F<em>y = ma</em>y$
   • Read the components of the vectors from the free-body diagram.
   • If needed, use kinematics to find positions and velocities.
3. ASSESS
   • Is the result reasonable?

QuickCheck Questions and Answers

• Question 3: Which of these objects is in equilibrium?
  • A. A car driving down the road at a constant speed
  • B. A block sitting at rest on a table
  • C. A skydiver falling at a constant speed
  • D. All of the above
• Question 4: Forces acting on a person hanging from a rope.
  1. Is this system in equilibrium?
  2. Represent the object of interest by a ’dot.’
  3. Identify all long-range forces.
  4. Identify all contact forces.
  5. Draw Force vectors showing the correct directions and relative magnitudes.
  6. If the weight of the person is 70 N, what is the tension on the rope? Object of interest: The person.
• Question 5 & 6: A steel beam hangs from a cable as a crane lifts the beam at a constant acceleration?
  • What forces act on the beam if the beam is moving up with a constant acceleration?
  • A. Gravity.
  • B. Gravity and tension in the cable.
  • C. Gravity and a force of motion.
  • D. Gravity and tension and a force of motion.
  • Is this system in equilibrium?
    • A. Yes
    • B. No
• Question 7: Can you draw the free body diagram if the beam is moving up with a constant acceleration?
• Question 8: An object on a rope is lowered at constant speed. Which of the following is true?
  • A. The rope tension is greater than the object’s weight.
  • B. The rope tension equals the object’s weight.
  • C. The rope tension is less than the object’s weight.
  • D. The rope tension can’t be compared to the object’s weight.
  • Hint: Draw a free body diagram first
• Question 9: An object on a rope is lowered at a steadily decreasing speed. Which is true?
  • A. The rope tension is greater than the object’s weight.
  • B. The rope tension equals the object’s weight.
  • C. The rope tension is less than the object’s weight.
  • D. The rope tension can’t be compared to the object’s weight.
  • Hint: Draw a free body diagram first
• Question 10: A ball has been tossed straight up. Which is the correct free-body diagram just after the ball has left the hand? Ignore air resistance.
• Question 11: A car is towed to the right at constant speed. Which is the correct free-body diagram?
• Question 12, 13 & 14: A box is resting on a smooth frictionless horizontal surface of a table.
  1. Is this system in equilibrium?
  2. Net force =?
• Question 15: A box is sitting on the floor of an elevator. The elevator is accelerating upward. The magnitude of the normal force on the box is
  • A. $F_N > mg$.
  • B. $F_N = mg$.
  • C. F_N < mg.
  • D. $F_N = 0$.
  • E. Not enough information to tell.
• Question 16: The top block is accelerated across a frictionless table by the falling mass m. The string is massless, and the pulley is both massless and frictionless. The tension in the string is
  • A. $F_T < mg$
  • B. $F_T = mg$
  • C. F_T > mg
• Question 17: How are acceleration of Block A and Block B related?
  • A. $a<em>{Ay} = a</em>{By}$
  • B. ${-a}<em>{Ay} = {-a}</em>{By}$
  • C. $a<em>{Ay} = {-a}</em>{By}$
  • D. $a<em>{By} = {-a}</em>{Ay}$
  • E. Either C or D.
• Question 18: A box is being pulled to the right over a rough surface. $F<em>T > F</em>{fs}$, so the box is speeding up. Suddenly the rope breaks. What happens? The box ___
  • A. Stops immediately.
  • B. Continues with the speed it had when the rope broke.
  • C. Continues speeding up for a short while, then slows and stops.
  • D. Keeps its speed for a short while, then slows and stops.
  • E. Slows steadily until it stops.
• Question 19: A box on a rough surface is pulled by a horizontal rope with tension $F_T$. The box is not moving. In this situation,
  • A. $F<em>{fs} = F</em>T$
  • B. $F<em>{fs} < F</em>T$
  • C. $F<em>{fs} > F</em>T$
  • D. $F_{fs} = 0$
• Question 20: A box is being pulled to the right at steady speed on a rough surface by a rope that angles upward. In this situation:
  • A. F_N > mg
  • B. $F_N = mg$
  • C. $F_N < mg$
  • D. $F_N = 0$
  • E. Not enough information to judge the size of the normal force.
• Question 21: A box with a weight of 100 N is at rest. It is then pulled by a 30 N horizontal force. Does the box move?
  • A. Yes
  • B. No

Alternative Notations

• Normal force:
  • $F_N = N = normal \ force$
• Gravitational force on an object:
  • $F_G = weight (W) = mg$
• Force of static friction:
  • $F<em>{fs} = f</em>s= \nu<em>s F</em>N$
• Force of kinetic friction:
  • $F<em>{fk} = f</em>k = \nu<em>k F</em>N$

Object on an Incline with Friction

1. Is this system in equilibrium?
2. Drawing the free bdy diagram:
   • Define your coordinate system
   • Represent the object of interest by a ’dot.’
   • Identify all long-range forces.
   • Identify all contact forces.
   • Draw Force vectors showing the correct directions. Note: For each force vector start drawing the vector from the dot.
3. Write down an expression for the net force on this person along the x-direction.
4. Write down an expression for the net force on this person along the y-direction.
5. Write down an expression for the acceleration of this person.
6. Question: A 100 kg block is sitting on an incline plane whose angle of inclination is 10 degrees. Find:
   • a) the parallel and perpendicular components of the force due to gravity.
   • b) the normal force.
   • c) the force of friction that keeps the block from sliding.
   • Note: if the box is at rest or moving at constant velocity along a particular axis, net force is zero along that axis. If it is speeding up or slowing down, then net force is ma.

Summary

• All forces arise from interaction between objects.
• The type of force (gravity, friction, normal force, magnetic force etc) depend on the nature of the interaction and sometimes on the nature of the objects themselves.
• We learnt how to draw free body diagrams (diagram showing forces acting on a body) and how to denote different kinds of forces.