Mechanics and Beam Problems

Problem 1: Beam with Uniform Loading

• A beam with a uniform loading of 'w' kg/m is supported at one end (A) and 2m from the other end (B).
• The reaction at support A is [wx^2]/[2(x-2)] kg.

Problem 2: Balance on a See-Saw

• When one boy sits 1.2m from the center, another boy must sit 1.5m from the center to balance.
• If the first boy carries an additional 14kg weight and sits 1.8m from the center, the second boy must move to 3m from the center.
• The weight of the heavier boy is 42kg.

Problem 3: Block on an Inclined Plane

• A 40kg block rests on an inclined plane at 20° from the horizontal with a coefficient of friction of 0.60.
• The force parallel to the incline required to cause impending motion down the plane is 77.

Problem 4: Block on an Inclined Plane with Friction

• A 250lb block is on a flat surface inclined at 30° with coefficients of kinetic and static friction of 0.30 and 0.40, respectively.
• The force required to start the block moving up the plane is 125lb.

Problem 5: Block on an Inclined Plane with Friction and Horizontal Force

• A 600N block rests on a surface inclined at 30° with an angle of friction of 15°.
• The horizontal force P required to prevent the block from sliding down is 164.60N.

Problem 6: Resultant Force Vector

• Three force vectors F1 = 4i + 2j + 5k, F2 = -2i + 7j - 3k, and F3 = 2i - j + 6k intersect at a single point.
• The magnitude of the resultant force vector R is 13.

Problem 7: Cable Suspension

• A cable is suspended between two supports at the same elevation, 500ft apart, with a load of 500lbs per horizontal foot, including the weight of the cable.
• The sag of the cable is 30ft, and the total length of the cable is 503.76ft.

Problem 8: Cable with Unit Weight

• A cable supported at two points of the same level has a unit weight of 0.02 kg per meter of horizontal distance.

Problem 1: Beam with Uniform Loading

• A beam with a uniform loading of 'w' kg/m is supported at one end (A) and 2m from the other end (B).
• The reaction at support A is [wx^2]/[2(x-2)] kg.

Problem 2: Balance on a See-Saw

• When one boy sits 1.2m from the center, another boy must sit 1.5m from the center to balance.
• If the first boy carries an additional 14kg weight and sits 1.8m from the center, the second boy must move to 3m from the center.
• The weight of the heavier boy is 42kg.

Problem 3: Block on an Inclined Plane

• A 40kg block rests on an inclined plane at 20° from the horizontal with a coefficient of friction of 0.60.
• The force parallel to the incline required to cause impending motion down the plane is 77.

Problem 4: Block on an Inclined Plane with Friction

• A 250lb block is on a flat surface inclined at 30° with coefficients of kinetic and static friction of 0.30 and 0.40, respectively.
• The force required to start the block moving up the plane is 125lb.

Problem 5: Block on an Inclined Plane with Friction and Horizontal Force

• A 600N block rests on a surface inclined at 30° with an angle of friction of 15°.
• The horizontal force P required to prevent the block from sliding down is 164.60N.

Problem 6: Resultant Force Vector

• Three force vectors F1 = 4i + 2j + 5k, F2 = -2i + 7j - 3k, and F3 = 2i - j + 6k intersect at a single point.
• The magnitude of the resultant force vector R is 13.

Problem 7: Cable Suspension

• A cable is suspended between two supports at the same elevation, 500ft apart, with a load of 500lbs per horizontal foot, including the weight of the cable.
• The sag of the cable is 30ft, and the total length of the cable is 503.76ft.

Problem 8: Cable with Unit Weight

• A cable supported at two points of the same level has a unit weight of 0.02 kg per meter of horizontal distance.

Definition of Rigid Body Dynamics

• Rigid body dynamics studies the motion of rigid objects, which maintain their shape and size during motion.

Assumptions of Rigid Body Dynamics

• Objects are continuous and unbroken.
• Objects have a fixed shape and size.
• Objects are not subject to deformation or change in shape.

Types of Motion in Rigid Body Dynamics

Translation

• Motion of the object as a whole, where every point moves in the same direction and by the same amount.

Rotation

• Motion of the object around a fixed axis, where every point moves in a circular path.

Kinematics of Rigid Body Motion

Describing Motion

• Position: described by the position of a reference point (e.g., center of mass).
• Orientation: described by the orientation of the object in space (e.g., using Euler angles or quaternions).
• Velocity: described by linear velocity of the reference point and angular velocity around the axis of rotation.
• Acceleration: described by linear acceleration of the reference point and angular acceleration around the axis of rotation.

Kinetics of Rigid Body Motion

Forces and Torques

• Force: causes linear acceleration of the object.
• Torque: causes angular acceleration of the object.

Newton's Second Law and Euler's Equations

• Newton's Second Law: F = ma, where F is the net force, m is the mass, and a is the acceleration.
• Euler's Equations: describe the rotation of a rigid body in terms of torque and angular velocity.

Angular Momentum

Definition and Conservation

• Angular momentum: product of moment of inertia, angular velocity, and distance from the axis of rotation.
• Conservation: total angular momentum of a closed system remains constant, unless acted upon by an external torque.

Energy of Rigid Body Motion

Kinetic and Potential Energy

• Kinetic energy: energy of motion, described by linear and angular velocity.
• Potential energy: energy of position, described by height or configuration of the object.

Applications of Rigid Body Dynamics

Fields of Application

• Robotics: models and controls motion of robots.
• Computer Graphics: simulates motion of objects in virtual environments.
• Mechanical Engineering: designs and analyzes mechanical systems, such as gears and linkages.