बल का क्षण – समस्याएं और समाधान

बल का क्षण – समस्याएं और समाधान

1. If F_R is the net force of F₁, एफ₂, और एफ₃, what is the magnitude of force F₂ and x?

ज्ञात :

Net force (F_R) = 40 एन

बल 1 (एफ₁) = 10 एन

Force (F₃) = 20 एन

चाहता था: The magnitude of force F₂ and distance of x

समाधान:

Find the magnitude of force F₂ :

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

ΣF = 0

- एफ_R + एफ₁ + एफ₂ - एफ₃ = 0

– 40 + 10 + F₂ - 20 = 0

– 30 + F₂ - 20 = 0

– 50 + F₂ = 0

F₂ = 50 न्यूटन।

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

Find x.

Choose A as the axis of rotation.

τ₁ = एफ₁ l₁ = (10 N)(1 m) = 10 Nm

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

τ₂ = एफ₂ x = (50)(x) = 50x Nm

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

τ₃ = एफ₃ 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 बल का क्षण :

Στ = 0

10 + 50x – 35 = 0

50x - 25 = 0

50x = 25

एक्स = एक्सएनएनएक्स / एक्सएनएनएक्स

x = 0.5 m

2. Forces of F₁, एफ₂, एफ₃, और एफ₄ 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.

ज्ञात :

बल एफ₁ = 10 N, the lever arm l₁ = 0

बल एफ₂ = 4 N, the lever arm l₂ = 2 मीटर

बल एफ₃ = 5 N, the lever arm l₃ = 3 मीटर

बल एफ₄ = 10 N, the lever arm l₄ = 6 मीटर

वांछित : the moment of force about point A

समाधान:

Moment of force 1 (τ₁) = एफ₁ l₁ = (10)(0) = 0

Moment of force 2 (τ₂) = एफ₂ l₂ = (4)(2) = -8 Nm

Moment of force 3 (τ₃) = एफ₃ l₃ = (5)(3) = 15 Nm

Moment of force 4 (τ₄) = एफ₄ 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.

बल आघूर्ण का परिणामी:

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 = एफ_C = 10 एन और एफ_B = 20 N, as shown in figure below. If distance of AB = BC = 20 cm, what is the moment of force about point C.

ज्ञात :

The axis rotation at point C.

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 सेमी

F_A = 10 न्यूटन

F_B = 20 न्यूटन

F_C = 10 न्यूटन

वांछित : The resultant of the moment of force about point C.

समाधान:

Moment of force A :

अनुसूचित जनजाति_A = (एफ_A)(आर_AC पाप 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 :

अनुसूचित जनजाति_B = (एफ_B)(आर_BC पाप 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 :

अनुसूचित जनजाति_C = (एफ_C)(आर_CC पाप 90^o) = (10 N)(0)(1) = 0

बल आघूर्ण का परिणामी:

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.

ज्ञात :

The axis rotation at point C.

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

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 न्यूटन

F₂ = 10 न्यूटन

F₃ = 10 न्यूटन

वांछित : Resultant of moment of force about point C.

समाधान:

Moment of force 1 :

अनुसूचित जनजाति₁ = (एफ₁)(आर₁ पाप 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 :

अनुसूचित जनजाति₂ = (एफ₂)(आर₂ पाप 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 :

अनुसूचित जनजाति₃ = (एफ₃)(आर₃ पाप 30^o) = (10 N)(0,2 m)(0,5) = -1 N.m

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

बल आघूर्ण का परिणामी:

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, एफ₂, और एफ₃ act on a rod as shown in figure below. Length of rod is 4 meters. What is the moment of force about point C.

(पाप 53^o = 0.8, cos 53^o = 0.6, AB = BC = CD = DE = 1 meter)

ज्ञात :

The axis of rotation at point C.

बल 1 (एफ₁) = 5 न्यूटन

F की क्रिया रेखा के बीच की दूरी₁ घूर्णन अक्ष के साथ (r₁) = 2 मीटर

बल 2 (एफ₂) = 0.4 न्यूटन

F की क्रिया रेखा के बीच की दूरी₂ घूर्णन अक्ष के साथ (r₂) = 1 मीटर

बल 3 (एफ₃) = 4.8 न्यूटन

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

चाहता था: The moment of force about point C.

समाधान:

Moment of force 1 :

τ₁ = एफ₁ 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.

Moment of force 2 :

τ₂ = एफ₂ 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 = एफ₃ r sin 90^o = (4,8 N)(2 m)(1) = -9,6 N

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

बल आघूर्ण का परिणामी:

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?

ज्ञात :

बिंदु O पर घूर्णन अक्ष।

बल 1 (एफ₁) = 6 न्यूटन

F की क्रिया रेखा के बीच की दूरी₁ घूर्णन अक्ष के साथ (r₁) = 1 मीटर

बल 2 (एफ₂) = 6 न्यूटन

F की क्रिया रेखा के बीच की दूरी₂ घूर्णन अक्ष के साथ (r₂) = 2 मीटर

बल 3 (एफ₃) = 4 न्यूटन

F की क्रिया रेखा के बीच की दूरी₃ घूर्णन अक्ष के साथ (r₃) = 2 मीटर

चाहता था: The resultant of the moment of force about point C

समाधान:

Moment of force 1 :

τ₁ = एफ₁ l₁ = (6 N)(1 m) = 6 Nm

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

Moment of force 2 :

τ₂ = एफ₂ r₂ पाप 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 = एफ₃ l₃ = (4 N)(2 m) = -8 Nm

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

बल आघूर्ण का परिणामी:

Στ = τ₁ + τ₂ – τ₃ = 6 + 6 – 8 = 4 N.m

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