H2 Physics 2 - Forces and Moments: Equilibrium of a Beam

Step 1 - Set up the problem

A uniform beam of length 2 m and weight 40 N rests horizontally on two supports.

Step 1 - Set up the problem

Support A is at the left end; support B is 0.5 m from the right end.

Step 1 - Set up the problem

A 60 N load hangs from the right end. Find the reaction forces at A and B.

Step 2 - Apply the condition for translational equilibrium

For equilibrium, total upward force equals total downward force.

Step 2 - Apply the condition for translational equilibrium

R_A plus R_B equals 40 plus 60, giving 100 N.

Step 2 - Apply the condition for translational equilibrium

One equation, two unknowns - we need to take moments.

Step 3 - Take moments about support A

Taking moments about A removes R_A from the equation.

Step 3 - Take moments about support A

Clockwise: 40 N at 1 m, and 60 N at 2 m.

Step 3 - Take moments about support A

Anticlockwise: R_B at 1.5 m.

Step 3 - Take moments about support A

So 40(1) + 60(2) = R_B(1.5).

Step 4 - Solve for both reaction forces

Dividing by 1.5: R_B = 107 N (to 3 s.f.).

Step 4 - Solve for both reaction forces

From equation (1): R_A = 100 minus 107 = negative 6.7 N.

Step 4 - Solve for both reaction forces

The negative sign means support A pushes downward - the beam lifts at that end.

Step 5 - Verify with moments about B

To verify, take moments about B.

Step 5 - Verify with moments about B

Since R_A acts downward, its clockwise moment about B is 6.7 times 1.5 = 10 N m.

Step 5 - Verify with moments about B

The beam weight gives a clockwise moment of 40 times 0.5 = 20 N m.

Step 5 - Verify with moments about B

The load gives an anticlockwise moment of 60 times 0.5 = 30 N m. Clockwise total: 10 + 20 = 30. This matches, confirming our answer.

Step 5 - Verify with moments about B

Final answers: R_B is approximately 107 N upward; R_A is approximately 6.7 N downward.