Hai hệ lực được gọi là tương đương khi:

Hợp của hệ lực là:

Tác dụng của một lực lên vật rắn không đổi nếu:

Liên kết là:

Phản lực liên kết là:

Hệ lực đồng quy là:

Hệ lực song song là:

Ngẫu lực là:

Hai ngẫu lực tương đương nhau là:

Trong hình vẽ, lực tác dụng trên đoạn AB là loại lực gì?

Trong hình vẽ, lực tác dụng tại điểm B là loại lực gì?

A simply supported beam has a uniform distributed load of 5 kN/m on segment AB (2 m) and a concentrated load of 10 kN at point C, 1 m beyond B. Which classification best describes the loading on segment AB?

On a beam AB, the distributed load increases linearly from 4 kN/m at A to 8 kN/m at B across span AC (3 m). What is the correct classification of the load acting on segment AC?

On the same beam, consider segment CB (3 m) where the distributed load continues increasing linearly from its value at C to 8 kN/m at B. Which classification best fits the load on segment CB?

A cylindrical pin at support B permits rotation but prevents translation in the plane. Which type of connection does this represent?

In Figure 1.1, what is the support at A of the beam AB under a uniform load p = qa?

In Figure 1.1, what is the support at C of the beam when the right end allows horizontal movement but resists vertical motion?

In Figure 1.2, identify the support at A where the beam end is rigidly embedded into a wall and resists rotation.

For segment AB in the diagram with 5 kN/m uniformly distributed over AB and a 10 kN point load at C, what type of loading acts on AB?

In the final diagram, which type of support is shown at B if it prevents rotation and translation in all directions?

In the diagram, a straight bar CD is connected at point C to a fixed corner and supports a small roller at D. A cable from the upper left pulls at C, and a horizontal member CB with a roller at B carries a vertical load P. For the bar CD, what is the direction of its reaction force at joint C?

In the L‑shaped frame shown, a vertical member AB is fixed at A, a horizontal member CD is attached by a pin at C, and the lower right corner E sits on a small roller on a horizontal track. What type of support exists at point E?

Consider a beam AB resting on supports with a vertical link at C connected to an inclined bar CD that is clamped to a wall at D. At point C, the beam connects to the inclined bar by a pin. What type of connection is at C?

In the frame shown, bar BD forms a 60° angle with the horizontal x-axis. If the reaction at B on BD is along the member AB which is horizontal, what is the angle between the reaction of bar BD and the x-axis?

For the same frame, determine the angle between the reaction of bar BD and the vertical y-axis when BD is inclined at 60° to the x-axis.

In the L-shaped member supported at point E by a slot along the horizontal line AE, what is the direction of the reaction at E? Assume the slot constrains motion only perpendicular to AE.

A beam is supported at A and C with a uniformly distributed load of 5 kN/m on the left span and a 10 kN point load near C. At A, the support prevents both translation and rotation. What type of connection is at A?

In the figure, joint A attaches a beam to a wall with no rotation allowed and all translations restrained. Which support best describes joint A?

A shaft passes through a bracket at A using a pin that allows rotation but prevents translation of the pin center. Which connection type is at A?

A simply supported beam rests on a support at A that prevents vertical translation but allows horizontal movement and rotation. Which support is modeled at A?

Location B uses a support that restrains both horizontal and vertical translation while allowing rotation about the support. Which designation best matches support B?

A uniform distributed load q acts over the middle 2a segment of beam AB of total length 4a. The equivalent resultant force P equals qa and acts at the segment’s center. What is the moment of P about point A, taking counterclockwise as positive?

A beam AB of length 4a carries a uniform load q from A to B. The resultant force is P=qa acting at distance 2a from A. What is the moment of the distributed load q about point A (counterclockwise positive)?

Beam ABC has segments AB=a and BC=2a. A uniform load q acts only over BC. Its resultant P=qa acts at the centroid of BC, a distance a from C toward B. What is the moment of P about point C, taking counterclockwise as positive?

A prismatic beam AB of length 4a is supported at A and B, with a uniformly distributed load q acting downward on the right half segment BC of length 2a. Point C is at the right end. Determine the moment of the distributed load q about point C. Take clockwise as negative.

A cantilever beam ABC fixed at A has total length 3a. A concentrated load P acts downward at point B located at distance a from A, and a uniformly distributed load q acts downward on the free end segment BC of length 2a. Determine the moment of the force P about point A. Take counterclockwise as positive.

A prismatic beam ABC of length 3a is fixed at A and carries a uniformly distributed load q downward over the entire span BC of length 2a to the right of A. Determine the moment of the distributed load q about point A. Take clockwise as negative.

A straight bar ABC lies along the x-axis. Two vertical forces act in the Az direction: 5 kN upward at a point 2 m from A, and 10 kN downward at point C which is 3 m from A. Determine the reaction force at A in the Az direction (positive upward) assuming only vertical forces and supports along z.

A beam AB is supported and loaded as shown: a vertical force F acts downward at a point B a distance a from A. Determine the moment of force F with respect to point A. Take counterclockwise as positive.

For the beam shown, a force F = 500 N acts at point A with perpendicular distance 3 m to point B and 2 m to point S along the beam as indicated. Determine the moment of F about point A. Choose the correct value and sign convention shown.

Beam ABC is loaded and supported as shown. Given F = 8 N, compute the moment of force F about point A.

Beam ABC with member CD is loaded as shown. Let S be the internal force in member CD. What is the moment of force S about point A? Distances AB and AC are a (m).

Let S1 be the internal force in bar AB as shown. What is the moment of force S1 about point E?

A force S acts along bar BD at 60° above the x-axis. Point C lies a distance a from point B along the x-axis. What is the moment of S about point C?

For the same bar-and-force system, what is the moment of the force S about point B? Assume the line of action of S passes through B.

A couple of magnitude M acts in the plane of the bar system with axis perpendicular to the plane. What is the moment of this couple about point B?

At a pin joint A in three-dimensional space with coordinate axes Axyz, how many independent reaction force components can the support exert?

A prismatic beam ABC is loaded on segment AC by a linearly varying distributed load from 4 kN/m at A to 8 kN/m at C over length 3 m. Taking point A as reference, what is the net moment of this distributed load system about A? Use clockwise negative convention.

Two bars AB and CD are connected as shown. Let S1 and S2 be the internal axial forces in bars AB and CD, respectively. Considering equilibrium along axis x, what is the sum of the projections of all forces on x for the pin-jointed linkage BEK?

A shaft carries two forces F1=500 N and F2=300 N acting at radii r1=80 mm and r2=300 mm, respectively, as shown. What is the total moment of F1 and F2 about the z-axis? Use N·mm units and right-hand rule.

For three concurrent forces T1, T2, and Q acting at point A, the x-axis is to the right. Force T1 makes 45° above the negative x-axis, T2 acts upward along the positive y-axis, and Q acts downward along the negative y-axis. Which expression gives the sum of projections of these forces on the x-axis?

A rigid bar ABC is acted on by three forces T1 at A horizontal to the left, T2 at A vertical upward, and T3 at C making 30° above the positive x-axis to the right. What is the algebraic sum of the projections of T1, T2, T3 onto the x-axis? Take rightward components as positive.

When the reference point for the reduced system of forces is changed (i.e., you choose a new reduction pole), how does the principal vector of the force system change?

Calling R the resultant vector of a force system reduced at point A. When changing the reduction point from A to B, the principal moment changes by an amount equal to:

Let the principal vector and principal moment of the force system be R̄ and M̄. The general equilibrium conditions of a force system are:

For a rigid body in space under a general force system, how many equations can be written to determine the equilibrium conditions in the Oxyz coordinate system?

A rigid body under coplanar forces in the Oxy plane requires how many equations to determine equilibrium conditions?

A rigid body under a system of parallel forces in space requires how many equations to determine equilibrium conditions in Oxyz?

A rigid body under a concurrent force system in space requires how many equations to determine equilibrium conditions in Oxyz?

For a rigid body under a concurrent force system in the Oxy plane, how many independent equilibrium equations are required to determine balance?

When shifting the force system’s reduced point from O to A, how does the scalar product between the principal resultant vector and the principal moment vector change?

A force system is reduced to a wrench when the principal vectors R̄ and M̄ satisfy which condition?

A uniform beam AB of length 4a is subjected to a downward distributed load q over segment BC of length 2a from the midpoint, a point load P at location a from A, and support reactions Y (vertical at A) and N (vertical at C). Which Ay-direction equilibrium equation is correct?

For the same beam, taking moments about point A (counterclockwise positive), which moment equilibrium expression is correct if the distributed load over 2a has its resultant q·2a acting at its centroid located at distance 3a from A, and point load P acts at distance a from A, with reaction N at C at distance 4a from A?

For the beam ABC fixed at A with distributed load q over length 2a and a point force P at distance a from A, which equilibrium equation along the Ay direction is correct?

For the same beam, which moment equilibrium equation about point A is correct if a clockwise positive sign convention is used?

A beam AB is supported by reactions Y at A and N at C, with a uniform load q over 3a and a point force P at A. What is the correct moment equilibrium about point C?

For a simply supported beam AB with reactions Y at A and N at B under a uniform load q over length 3a, what is the value of reaction N when the beam is in equilibrium?

For the beam AB with a uniform load q over length 4a and a point load N at the right end (as shown), the vertical reaction at support A is Y. What is the value of Y for equilibrium of AB?

Consider a cantilever bar ABC fixed at A with distributed load q over segment BC of length 2a and a point load P = qa at B. Taking moments about point B, which equation expresses moment equilibrium?

For the cantilever ABC fixed at A with the same loading, determine the reaction moment M at A when segment AC is in equilibrium. Assume P = qa and distributed load q acts over length 2a.

A beam AB is supported with reaction components X, Y, T and subjected to an applied force F at an angle. Which equation represents force equilibrium along the x-direction (Ax)?

For the bar AB with reaction components X, Y, T and an applied force F as shown, what is the correct moment-equilibrium equation about point B?

For the same bar, which equation represents the moment equilibrium about point A?

Given the bar AB with reactions X, Y, T and force F = 500 N as shown, what is the value of the reaction Y when the system is in equilibrium?

A rigid bar AB is supported at A by pin reactions X and Y, and at C by a link BC carrying axial force T along BC at 30° above the horizontal. A vertical force F = 500 N acts downward at midspan between A and B. Distances: A–midpoint = 2 m, midpoint–B = 2 m, and C is at the right end. When the bar is in equilibrium, what is the value of the reaction T in link BC?

For the same bar AB with reactions X, Y at A and a support force S at D acting along a 45° direction to the negative x-axis, a vertical force F = 500 N acts downward at the midpoint. Write the correct force equilibrium equation along the x-axis.

Consider bar AB with pin reactions X, Y at A and an inclined support force S at D, with a downward force F = 500 N acting at a point 2 m from A and 2 m from B. Taking moments about point C located 2 m from B toward A, which equilibrium moment equation is correct?

For the configuration with reactions X, Y at A, an inclined support S at D, and a downward force F = 500 N at the midpoint, what is the vertical reaction Y at support A when the bar is in equilibrium?

Bar AB is supported at A and B with reaction components X and Y at B and A, and an inclined reaction S at point near A as shown. A horizontal force F = 500 N is applied at B. Distances are: A to load point C is 2 m, C to B is 2 m, and the vertical height of A above the lower reference is 2 m. When the system is in equilibrium, the magnitude of the reaction S is:

For the same bar AB with reactions X, Y, S and applied force F = 500 N at B, write the scalar moment equilibrium equation about point A using the forces shown. Choose the correct expression:

Member BDEK has support reactions S1, S2, and N and external forces as shown. What is the correct equilibrium equation for the sum of forces in the vertical direction, ΣEy = 0?

For the vertical member BDEK, the support reactions are S1 at B, S2 at D, and a normal reaction N at the guide near E. A horizontal force F acts at K on the right, at height a above E, and the link distances are shown with a square layout of side a. Writing the moment equilibrium about point E (taking counterclockwise positive), which expression is correct?

Considering the same frame, write the moment equilibrium about point D for the forces S1 at B, S2 at D, the normal reaction N near E, and a horizontal force F applied at K at a vertical distance a above E. Which relation satisfies ΣM_D = 0 with counterclockwise positive?

In the planar structure BDEK, the external force is F = 100 N acting horizontally to the right at point D. The bar is supported by a pin at E, producing a reaction with horizontal component N only. When the system is in equilibrium, what is the value of reaction N at E?

Consider a rigid bar AB of length 2 m connected to the frame BDEK by pins at A and B. A horizontal force F = 100 N acts at D to the right. Reaction forces at A are S1 (horizontal) and S2 (vertical). When the system is in equilibrium, what is the value of S1 at A?

For the same system as above, with F = 100 N to the right and supports providing components S1 and S2 at A, determine the vertical reaction S2 at A when the system is in equilibrium.

For the beam ABC with unknown reaction components S, Xc, Yc and a force S inclined at 60° to the positive x-axis at point D as shown, the equilibrium equation for the projection of forces on axis Bx is:

For the same beam, the equilibrium equation for the projection of forces on axis By is:

Taking moments about point C for all external forces on the beam, the correct scalar moment equilibrium equation is:

Given M = 5 kN·m, q = 2 kN/m, and a = 2 m for the beam shown, the magnitude of the inclined reaction S at member BD that satisfies equilibrium is:

With the same data (M = 5 kN·m, q = 2 kN/m, a = 2 m), the horizontal reaction at C is:

With the same data, the vertical reaction at C is:

For beam ACB with vertical reaction Y at A and vertical reaction N at B, carrying a uniformly distributed load q1 = 4 kN/m over the 3 m span AC, which correct force projection equilibrium along Ay is satisfied? Assume upward positive and the load acts downward over 3 m.

For the same beam, taking moments about point A with counterclockwise positive, which moment equilibrium equation is correct? The resultant of the distributed load is q1·3 acting at 1.5 m from A.

Using equilibrium, what is the reaction force N at support B? Take the span AB = 6 m and the distributed load q1 = 4 kN/m over the first 3 m from A.

What is the reaction force Y at support A for the same loading? Use vertical force and moment equilibrium with q1 = 4 kN/m on 3 m and span AB = 6 m.

For the force system at point A with forces T1 at 45° above +x, T2 at 30° above −x, and a downward force Q, the equilibrium equation along Ax is:

For the same system, the equilibrium equation along Ay is:

Given Q = 300 N acting downward and the same directions for T1 (45°) and T2 (30°), the equilibrium yields the value of T1 as:

For a beam ABC fixed at A, with forces T1 vertical at AB, T2 inclined left at 45° at A, and T3 along BC at 30° above +x, the Ax equilibrium equation using support reaction X at A is:

For the same beam, the Ay equilibrium equation with vertical reaction Y at A is:

Which statement correctly describes the role of reaction moment M at a fixed support in planar equilibrium for the beam ABC?

In 3D space, a point M moves with x = x(t), y = y(t), z = z(t). Which expression gives the magnitude of the total velocity v_M?

In 3D space, a point M moves with x = x(t), y = y(t), z = z(t). Which expression gives the magnitude of the total acceleration a_M?

Which statement is correct about differentiating the position vector r(t) of a moving point?

Given x = x(t), y = y(t), z = z(t), what is this set of equations called in Cartesian coordinates?

If you eliminate t from x = x(t), y = y(t), z = z(t) to relate x, y, z, what do you obtain?

In natural coordinates, the total acceleration of a point M is given by which expression, where v is speed, ρ is radius of curvature, t̂ is the unit tangent, and n̂ is the unit normal?

For motion in natural coordinates, the correct expression for the normal (centripetal) component of acceleration is:

Which statement is correct about the normal component of acceleration in natural coordinates?

A point moves as x = 3 cos(4t), y = 3 sin(4t), z = 2t (meters, seconds). What is the trajectory of the point?

A point moves as x = 4 cos(2t), y = 3 sin(2t), z = 0 (meters, seconds). What is the trajectory of the point?

A point M moves on a circle of radius 6 cm with position given by $$s = 6t + 9t^2 + 12t^3$$ (cm), where s is arc length along the circle and t in seconds. What is the speed of M at t = 2 s?

A point moves in the plane with parametric equations x = 75 cos(4t^2), y = 75 sin(4t^2), with t in seconds and x,y in cm. What is the correct expression for the instantaneous speed v(t)?

Given the motion $$x = 75 \cos(4t^2)$$ , $$y = 75 \sin(4t^2)$$ (cm), which statement best describes the trajectory of the point?

A point M moves on a circle of radius 10 cm with arc-length function $$s = 4t^3 + 2t$$ (cm). What is the speed of M at t = 5 s?