Coplanar Force Systems Quiz

Force System Magnitude

• The resultant force of a system can be calculated using vector addition.
• It is important to consider both the magnitude and direction of each force when finding the resultant.
• This can be done graphically or using trigonometry.

Component of Weight

• The component of weight perpendicular to the inclined plane is the force that acts to keep the block from sliding down the plane.
• It can be calculated using trigonometry.
• This force is equal to the weight of the block multiplied by the cosine of the angle between the plane and the horizontal.

Inclination Angle

• Rope B should be inclined at an angle that maximizes the horizontal component of the force applied.
• This will allow for the most efficient pulling of the vehicle.
• The optimal angle can be determined using trigonometry.

Unknown Force

• The unknown force 'P' and its inclination with the x-axis can be found by resolving the resultant force into its horizontal and vertical components.
• The horizontal component of the resultant force must be equal to the sum of the horizontal components of all the forces acting on the system.
• The vertical component of the resultant force must be equal to the sum of the vertical components of all the forces acting on the system.

Varignon’s Theorem

• Varignon’s Theorem states that the moment of a force about a point is equal to the sum of the moments of the components of the force about the same point.
• This theorem is useful for simplifying calculations involving moments.

Proof of Varignon’s Theorem

• Consider a force F acting at a point A.
• Resolve this force into two components, Fx and Fy, acting along the x and y axes respectively.
• The moment of the force F about a point O is given by M = F * d, where d is the perpendicular distance from O to the line of action of F.
• The moment of Fx about O is given by Mx = Fx * dy, where dy is the perpendicular distance from O to the line of action of Fx.
• Similarly, the moment of Fy about O is given by My = Fy * dx, where dx is the perpendicular distance from O to the line of action of Fy.
• Adding Mx and My, we get M = Fx * dy + Fy * dx.
• Since Fx = F * cosθ and Fy = F * sinθ, where θ is the angle between F and the x-axis, we can write M = F * cosθ * dy + F * sinθ * dx.
• Using the trigonometric identity cosθ * dy + sinθ * dx = d, we get M = F * d.
• Since M = F * d, this proves that the moment of a force about a point is equal to the sum of the moments of the components of the force about the same point.

Test your knowledge of force systems with this practice quiz. Solve problems related to coplanar forces and determine unknown forces and their inclinations with respect to the x axis.