Moment sile – problemi i rješenja

Moment sile – problemi i rješenja

1. If F_R is the net force of F₁, F₂i F₃, what is the magnitude of force F₂ and x?

Poznato:

Net force (F_R) = 40 N

Force 1 (F₁) = 10 N

Force (F₃) = 20 N

Htjela: The magnitude of force F₂ and distance of x

rješenje:

Find the magnitude of force F₂ :

Force points to upward, signed negative and force points to downward, signed negative.

ΣF = 0

- F_R + F₁ + F₂ - F₃ = 0

– 40 + 10 + F₂ - 20 = 0

– 30 + F₂ - 20 = 0

– 50 + F₂ = 0

F₂ = 50 Newtona.

Plus sign indicates that the direction of the force is upward.

Find x.

Choose A as the axis of rotation.

τ₁ =F₁ l₁ = (10 N)(1 m) = 10 Nm

The torque 1 rotates beam counterclockwise so we assign positive sign to the torque 3.

τ₂ =F₂ x = (50)(x) = 50x Nm

The torque 1 rotates beam counterclockwise so we assign positive sign to the torque 3.

τ₃ =F₃ x = (20 N)(1.75 m) = -35 Nm

The torque 2 rotates beam clockwise so we assign negative sign to the torque 2.

The net of moment sile :

Στ = 0

10 + 50x – 35 = 0

50x - 25 = 0

50x = 25

x = 25/50

x = 0.5 m

2. Forces of F₁, F₂, F₃i F₄ acts on the rod of ABCD as shown in figure. If rod’s mass ignored, what is the magnitude of the moment of force, about point A.

The axis of rotation = points A.

Poznato:

Forsiraj F₁ = 10 N, the lever arm l₁ = 0

Forsiraj F₂ = 4 N, the lever arm l₂ = 2 metra

Forsiraj F₃ = 5 N, the lever arm l₃ = 3 metra

Forsiraj F₄ = 10 N, the lever arm l₄ = 6 metra

Traži se: the moment of force about point A

rješenje:

Moment of force 1 (τ₁) = F₁ l₁ = (10)(0) = 0

Moment of force 2 (τ₂) = F₂ l₂ = (4)(2) = -8 Nm

Moment of force 3 (τ₃) = F₃ l₃ = (5)(3) = 15 Nm

Moment of force 4 (τ₄) = F₄ l₄ = (10)(6) = -60 Nm

If torque rotates rod counterclockwise then we assign positive sign.

If torque rotates rod clockwise then we assign negative sign.

The resultant of the moment of force :

τ = 0 – 8 Nm + 15 Nm – 60 Nm

τ = -68 Nm + 15 Nm

τ = -53 Nm

Minus sign indicates that the moment of force rotates rod clockwise.

3. Three forces act on a rod, F_A =F_C = 10 N i F_B = 20 N, as shown in figure below. If distance of AB = BC = 20 cm, what is the moment of force about point C.

Poznato:

The axis rotation at point C.

Distance between F_A and the axis of rotation (r_AC) = 40 cm = 0,4 metra

Distance between F_B and the axis of rotation (r_BC) = 20 cm = 0.2 metra

Distance between F_C and the axis of rotation (r_CC) = 0 cm

F_A = 10 Newtona

F_B = 20 Newtona

F_C = 10 Newtona

Traži se: The resultant of the moment of force about point C.

rješenje:

Moment of force A :

Στ_A = (F_A)(r_AC grijeh 90^o) = (10 N)(0,4 m)(1) = -4 Nm

Minus sign indicates that the moment of force rotates rod clockwise.

Moment of force B :

Στ_B = (F_B)(r_BC grijeh 90^o) = (20 N)(0,2 m)(1) = 4 Nm

Plus sign indicates that the moment of force rotates rod counterclockwise.

Moment of force C :

Στ_C = (F_C)(r_CC grijeh 90^o) = (10 N)(0)(1) = 0

The resultant of the moment of force :

Στ = Στ₁ + Στ₂ + Στ₃

Στ = -4 + 4 + 0

Στ = 0 Nm

4. Length of a rod is 50 cm. Three forces act on the rod, as shown in figure below. If the axis of rotation is point C, what is the net of the moment of force.

Poznato:

The axis rotation at point C.

Distance between F₁ and the axis of rotation is (r₁) = 30 cm = 0,3 metra

Distance between F₂ and the axis of rotation (r₂) = 10 cm = 0,1 metra

Distance between F₃ and the axis of rotation (r₃) = 20 cm = 0,2 metra

F₁ = 10 Newtona

F₂ = 10 Newtona

F₃ = 10 Newtona

Traži se: Resultant of moment of force about point C.

rješenje:

Moment of force 1 :

Στ₁ = (F₁)(r₁ grijeh 90^o) = (10 N)(0,3 m)(1) = -3 Nm

Minus sign indicates that the moment of force rotates rod clockwise.

Moment of force 2 :

Στ₂ = (F₂)(r₂ grijeh 90^o) = (10 N)(0,1 m)(1) = 1 Nm

Plus sign indicates that the moment of force rotates rod counterclockwise.

Moment of force 3 :

Στ₃ = (F₃)(r₃ grijeh 30^o) = (10 N)(0,2 m)(0,5) = -1 Nm

Minus sign indicates that the moment of force rotates rod clockwise.

The resultant of the moment of force :

Στ = Στ₁ + Στ₂ + Στ₃

Στ = -3 + 1 – 1

Στ = -3 Nm

Minus sign indicates that the resultant of the moment of force rotates rod clockwise.

5. Three forces F₁, F₂i F₃ act on a rod as shown in figure below. Length of rod is 4 meters. What is the moment of force about point C.

(sin 53^o = 0.8, cos 53^o = 0.6, AB = BC = CD = DE = 1 metar)

Poznato:

The axis of rotation at point C.

Force 1 (F₁) = 5 Newtona

The distance between the line of action of F₁ with the axis of rotation (r₁) = 2 metara

Force 2 (F₂) = 0.4 Newtona

The distance between the line of action of F₂ with the axis of rotation (r₂) = 1 metra

Force 3 (F₃) = 4.8 Newtona

The distance between the line of action of F₃ with the axis of rotation (r₃) = 2 metar

Htjela: The moment of force about point C.

rješenje:

Moment of force 1 :

τ₁ =F₁ r grijeh 53^o = (5 N)(2 m)(0,8) = (10)(0,8) N = 8 N

Plus sign indicates that the moment of force rotates rod counterclockwise.

Moment of force 2 :

τ₂ =F₂ r grijeh 90^o = (0,4 N)(1 m)(1) = -0,4 N

Minus sign indicates that the moment of force rotates rod clockwise.

Moment of force 3 :

τ₃ =F₃ r grijeh 90^o = (4,8 N)(2 m)(1) = -9,6 N

Minus sign indicates that the moment of force rotates rod clockwise.

The resultant of the moment of force :

Στ = τ₁ – τ₂ – τ₃ = 8 – 0,4 – 9,6 = 8 – 10 = 2 N.m

Plus sign indicates that the moment of force rotates rod counterclockwise.

6. What is the resultant of the moment of force about the axis of rotation at point O by forces acts on the rod, as shown in the figure below?

Poznato:

The axis of rotation at point O.

Force 1 (F₁) = 6 Newtona

The distance between the line of action of F₁ with the axis of rotation (r₁) = 1 metra

Force 2 (F₂) = 6 Newtona

The distance between the line of action of F₂ with the axis of rotation (r₂) = 2 metara

Force 3 (F₃) = 4 Newtona

The distance between the line of action of F₃ with the axis of rotation (r₃) = 2 metara

Htjela: The resultant of the moment of force about point C

rješenje:

Moment of force 1 :

τ₁ =F₁ l₁ = (6 N)(1 m) = 6 Nm

Plus sign indicates that the moment of force rotates rod counterclockwise.

Moment of force 2 :

τ₂ =F₂ r₂ grijeh 30^o = (6 N)(2 m)(0,5)= 6 Nm

Plus sign indicates that the moment of force rotates rod counterclockwise.

Moment of force 3 :

τ₃ =F₃ l₃ = (4 N)(2 m) = -8 Nm

Minus sign indicates that the moment of force rotates rod clockwise.

The resultant of the moment of force :

Στ = τ₁ + τ₂ – τ₃ = 6 + 6 – 8 = 4 Nm

Plus sign indicates that the moment of force rotates rod counterclockwise.