Rotational Equilibrium Examples in Physics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rotational Equilibrium.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Rotational equilibrium is the condition in which the net torque on an object is zero, so its angular velocity does not change.

If the clockwise and counterclockwise twists balance, the object will not start spinning faster one way or the other.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Rotational Equilibrium asks students to choose the object, list external interactions, and reason from the resulting force or torque pattern.

Common stuck point: Students often know a formula related to rotational equilibrium but skip the recognition step: Have I isolated one system and listed the external forces or torques acting on it before applying a law? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Have I isolated one system and listed the external forces or torques acting on it before applying a law?

Worked Examples

Example 1

medium
A uniform 60 N60\text{ N} horizontal bar, 2 m2\text{ m} long, is supported by two cables at its ends. Find each cable tension.

Answer

T1=T2=30 NT_1 = T_2 = 30 \text{ N}

First step

1
Weight acts at center; by symmetry each cable supports half.

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Example 2

hard
A uniform 80 N80\text{ N} beam, 4 m4\text{ m} long, is hinged at a wall. A 40 N40\text{ N} load hangs 3 m3\text{ m} from the hinge. A vertical cable at the far end (4 m4\text{ m}) holds the beam horizontal. Find the cable tension.

Example 3

hard
A uniform 200 N200\text{ N} sign, 2 m2\text{ m} wide, hangs from two ropes, one at each top corner. A 50 N50\text{ N} object is attached 0.4 m0.4\text{ m} from the left corner. Find the tension in each rope.

Example 4

challenge
A uniform 120 N120\text{ N} ladder, 5 m5\text{ m} long, leans against a frictionless wall at 70∘70^\circ to the floor. A 600 N600\text{ N} painter stands 4 m4\text{ m} up the ladder (measured along the ladder). Find the friction force needed at the floor. Use cos⁡70∘≈0.342\cos 70^\circ \approx 0.342, sin⁡70∘≈0.940\sin 70^\circ \approx 0.940.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
An object is in rotational equilibrium. What is the net torque on it?

Example 2

easy
A force of 10 N10\,\text{N} acts perpendicular to a wrench 0.3 m0.3\,\text{m} from the pivot. Find the torque.

Example 3

easy
A clockwise torque of 6 N\cdotpm6\,\text{N·m} acts on a beam. For rotational equilibrium, what counterclockwise torque is needed?

Example 4

easy
A seesaw has a 20 N20\,\text{N} weight 1 m1\,\text{m} left of the pivot and a 10 N10\,\text{N} weight on the right. For balance, find the right weight's distance.

Example 5

easy
A force of 8 N8\,\text{N} produces a torque of 4 N\cdotpm4\,\text{N·m} (perpendicular). Find the lever arm.

Example 6

easy
Two equal 15 N15\,\text{N} weights sit equal distances on opposite sides of a pivot. Is the system in rotational equilibrium?

Example 7

easy
A door handle is placed far from the hinge rather than near it. Why?

Example 8

easy
A torque of 12 N\cdotpm12\,\text{N·m} CW and a torque of 12 N\cdotpm12\,\text{N·m} CCW act on a wheel. Find the net torque.

Example 9

medium
A 4 N weight is 0.5 m0.5\,\text{m} left of a pivot and a 6 N weight is 0.2 m0.2\,\text{m} right. Find the net torque (CCW positive).

Example 10

medium
A uniform 30 N30\,\text{N} bar, 2 m2\,\text{m} long, is pivoted at one end. Find the torque due to its weight about the pivot.

Example 11

medium
A 100 N weight hangs 0.4 m0.4\,\text{m} from a pivot. A force is applied 0.8 m0.8\,\text{m} on the other side. Find the force needed for equilibrium.

Example 12

medium
A force of 20 N20\,\text{N} acts at 30∘30^\circ to a 0.6 m0.6\,\text{m} lever arm. Find the torque about the pivot.

Example 13

medium
A meter stick balances at its center. A 2 N2\,\text{N} weight at the 20 cm20\,\text{cm} mark balances a weight at the 70 cm70\,\text{cm} mark. Find that weight (pivot at 50 cm50\,\text{cm}).

Example 14

medium
A uniform 40 N40\,\text{N} beam, 3 m3\,\text{m} long, rests on a pivot 1 m1\,\text{m} from the left end. What downward force at the left end keeps it balanced?

Example 15

medium
Two children, 25 kg25\,\text{kg} at 1.6 m1.6\,\text{m} left and 40 kg40\,\text{kg} on the right of a seesaw pivot, balance (g=10 m/s2g = 10\,\text{m/s}^2). Find the right child's distance.

Example 16

medium
A 50 N weight hangs 0.3 m0.3\,\text{m} from a pivot. A force is applied 0.6 m0.6\,\text{m} on the other side. Find the force needed for equilibrium.

Example 17

medium
A uniform 24 N24\,\text{N} bar, 3 m3\,\text{m} long, is pivoted at one end. Find the torque due to its weight about the pivot.

Example 18

challenge
A uniform 60 N60\,\text{N} beam, 4 m4\,\text{m} long, is hinged at the wall and held horizontal by a vertical cable at the far end. A 40 N40\,\text{N} load hangs at 3 m3\,\text{m} from the hinge. Find the cable tension.

Example 19

challenge
A 5 m uniform beam (100 N100\,\text{N}) rests on two supports, one 1 m1\,\text{m} from the left end and one 1 m1\,\text{m} from the right end. Find the force on the left support.

Example 20

challenge
A uniform 80 N80\,\text{N} ladder, 5 m5\,\text{m} long, leans against a frictionless wall at 60∘60^\circ to the floor. Find the friction force the floor must provide (cos⁡60∘=0.5\cos 60^\circ = 0.5, sin⁡60∘=0.866\sin 60^\circ = 0.866).

Example 21

easy
A force of 20 N20\text{ N} acts perpendicular to a wrench 0.25 m0.25\text{ m} from the pivot. Find the torque.

Example 22

easy
A clockwise torque of 9 N\cdotpm9\text{ N·m} acts on a beam. Find the counterclockwise torque needed for equilibrium.

Example 23

easy
A 30 N30\text{ N} weight sits 0.8 m0.8\text{ m} left of a pivot; a 24 N24\text{ N} weight on the right. Find the right weight's distance for balance.

Example 24

easy
A 12 N12\text{ N} force produces a 6 N\cdotpm6\text{ N·m} torque (perpendicular). Find the lever arm.

Example 25

easy
Two equal 20 N20\text{ N} weights sit on opposite sides of a pivot, each 0.4 m0.4\text{ m} from it. Find the net torque.

Example 26

easy
A torque of 15 N\cdotpm15\text{ N·m} CW and a torque of 7 N\cdotpm7\text{ N·m} CCW act on a wheel. Find net torque (CCW positive).

Example 27

medium
A 6 N weight is 0.3 m0.3\text{ m} left of pivot, an 8 N weight is 0.4 m0.4\text{ m} right. Find net torque (CCW positive).

Example 28

medium
A uniform 40 N40\text{ N} bar of length 2 m2\text{ m} is pivoted at one end. Find the torque due to its weight about the pivot.

Example 29

medium
A 120 N weight hangs 0.5 m0.5\text{ m} from a pivot. A force is applied 1.5 m1.5\text{ m} on the other side. Find the force needed for balance.

Example 30

medium
A 30 N30\text{ N} force acts at 60∘60^\circ to a 0.4 m0.4\text{ m} lever arm. Find the torque about the pivot.

Example 31

medium
A meter stick is pivoted at its 50 cm50\text{ cm} mark. A 4 N4\text{ N} weight sits at 30 cm30\text{ cm}. Find the weight that balances at the 80 cm80\text{ cm} mark.

Example 32

medium
A uniform 50 N50\text{ N} beam, 4 m4\text{ m} long, rests on a pivot 1 m1\text{ m} from its left end. What downward force at the left end keeps it balanced?

Example 33

medium
Two children, 30 kg30\text{ kg} at 1.5 m1.5\text{ m} left and 45 kg45\text{ kg} on the right of a seesaw pivot, balance (g=10 m/s2g = 10\text{ m/s}^2). Find the right child's distance.

Example 34

medium
A door is 0.9 m0.9\text{ m} wide. A 50 N50\text{ N} force is applied perpendicular at the handle (far edge). Find the torque about the hinges.

Example 35

medium
A uniform 90 N90\text{ N} bar, 1.5 m1.5\text{ m} long, is pivoted at one end. Find the torque due to its weight about the pivot.

Example 36

medium
Why is the pivot a useful choice for the torque axis in a static problem?

Example 37

medium
A meter stick balances at 50 cm50\text{ cm}. A 5 N5\text{ N} weight is at the 10 cm10\text{ cm} mark. Find the weight at 60 cm60\text{ cm} that balances it.

Example 38

medium
A uniform 50 N50\text{ N} shelf, 0.8 m0.8\text{ m} wide, is supported by two brackets at its ends. A 30 N30\text{ N} book sits 0.3 m0.3\text{ m} from the left bracket. Find the force on the left bracket.

Example 39

hard
A uniform 40 N40\text{ N} beam, 3 m3\text{ m} long, is hinged at the wall and held horizontal by a vertical cable at the far end. Find the cable tension.

Example 40

hard
A 4 m4\text{ m} uniform beam (80 N80\text{ N}) rests on two supports: one 0.5 m0.5\text{ m} from the left end and one 0.5 m0.5\text{ m} from the right end. Find the force on each support.

Example 41

hard
A 6 m6\text{ m} uniform beam (60 N60\text{ N}) rests on two supports, one at the left end and one at the right end. A 90 N90\text{ N} load sits 2 m2\text{ m} from the left end. Find the force on each support.

Example 42

hard
A uniform 100 N100\text{ N} ladder, 4 m4\text{ m} long, leans against a frictionless wall at 60∘60^\circ to the floor. Find the friction force the floor must provide (cos⁡60∘=0.5\cos 60^\circ = 0.5, sin⁡60∘=0.866\sin 60^\circ = 0.866).

Example 43

hard
A 3 m3\text{ m} uniform plank (40 N40\text{ N}) overhangs a table edge. The plank rests fully on the table left of the edge, except a portion overhangs. What is the maximum overhang distance before tipping?

Example 44

challenge
A uniform 80 N80\text{ N} rod, 1 m1\text{ m} long, is hinged at one end. A horizontal cable attached at the free end holds it at 30∘30^\circ above horizontal. Find the cable tension.
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Related Concepts

TorqueEquilibrium

Background Knowledge

These ideas may be useful before you work through the harder examples.

torqueequilibrium