Moment of force – problems and solutions

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

Mōhiotia:

Net force (F_R) = 40 N Moment of force – problems and solutions 1

Kaha 1 (F₁) = 10 N

Force (F₃) = 20 N

Hiahia: The magnitude of force F₂ and distance of x

Rongoā:

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 Ngā Newtoni.

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 of force :

Στ = 0

10 + 50x – 35 = 0

50x – 25 = 0

50x = 25

x = 25/50

x = 0.5 mita

2. Forces of F₁, F₂, F₃, me 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.

Mōhiotia:

Te kaha F₁= 10 N, the lever arm l₁ = 0 Moment of force – problems and solutions 2

Te kaha F₂= 4 N, the lever arm l₂ = 2 mita

Te kaha F₃= 5 N, the lever arm l₃ = 3 mita

Te kaha F₄= 10 N, the lever arm l₄ = 6 mita

E hiahiatia ana: the moment of force about point A

Rongoā:

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.

A hi'o atoa Two parallel conductors carrying currents – problems and solutions

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 me 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.

Mōhiotia:

The axis rotation at point C. Moment of force – problems and solutions 3

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

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

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

F_A = 10 Niutona

F_B = 20 Niutona

F_C = 10 Niutona

E hiahiatia ana: The resultant of the moment of force about point C.

Rongoā:

Moment of force A :

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

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

Moment of force B :

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

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

Moment of force C :

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

The resultant of the moment of force :

Στ = Στ₁ + Στ₂ + Στ₃

Στ = -4 + 4 + 0

Στ = 0 N.m

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.

Mōhiotia:

The axis rotation at point C. Moment of force – problems and solutions 4

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

A hi'o atoa Te hurihanga o ngā waeine – ngā raruraru me ngā otinga

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

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

F₁ = 10 Niutona

F₂ = 10 Niutona

F₃ = 10 Niutona

E hiahiatia ana: Resultant of moment of force about point C.

Rongoā:

Moment of force 1 :

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

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

Moment of force 2 :

Στ₂= (F₂)(r₂ hara 90^o) = (10 N)(0,1 m)(1) = 1 N.m

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

Moment of force 3 :

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

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 N.m

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

5. Three forces F₁, F₂, me 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 meter)

Mōhiotia:

The axis of rotation at point C.

Kaha 1 (F₁) = 5 Niutona

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

Kaha 2 (F₂) = 0.4 Niutona

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

Kaha 3 (F₃) = 4.8 Niutona

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

Hiahia: The moment of force about point C.

Rongoā:

Moment of force 1 :

τ₁=F₁ r sin 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.

A hi'o atoa Rolling motion - problems and solutions

Moment of force 2 :

τ₂=F₂ r sin 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 sin 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?

Mōhiotia:

The axis of rotation at point O.

Kaha 1 (F₁) = 6 Niutona

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

Kaha 2 (F₂) = 6 Niutona

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

Kaha 3 (F₃) = 4 Niutona

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

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

Rongoā:

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 hara 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 N.m

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