A uniform beam PQ of length 4.0\text{ m} and mass 20\text{ kg} is supported horizontally o… — JAMB UTME

A uniform beam $PQ$ of length $4.0\text{ m}$ and mass $20\text{ kg}$ is supported horizontally on a pivot at end $P$ and by a vertical wire attached at end $Q$ . A load of mass $30\text{ kg}$ is placed on the beam at a distance of $1.0\text{ m}$ from $P$ . Taking the acceleration due to gravity $g = 10\text{ m/s}^2$ , what is the tension in the vertical wire attached at $Q$ in newtons?

Cevap: 175 N

Cevap

The tension in the vertical wire attached at end

Q

175\text{ N}

By applying the principle of moments about the pivot at

P

, the sum of downward clockwise moments produced by the

30\text{ kg}

load (

300\text{ N} \times 1.0\text{ m} = 300\text{ N}\cdot\text{m}

) and the beam's center of gravity (

200\text{ N} \times 2.0\text{ m} = 400\text{ N}\cdot\text{m}

) equals

700\text{ N}\cdot\text{m}

. Equating this to the counterclockwise moment of the tension force (

T \times 4.0\text{ m}

) gives

T = 175\text{ N}

Adım Adım Çözüm

Determine forces and their perpendicular distances from the pivot at

P

The load exerts a downward force of

300\text{ N}

1.0\text{ m}

from

P

. The uniform beam's weight of

200\text{ N}

acts at its midpoint (

2.0\text{ m}

from

P

). The vertical tension

T

acts upward at

Q

(

4.0\text{ m}

from

P

Before applying the principle of moments, all force magnitudes and their distance arms relative to the pivot point must be identified.

Equate total clockwise moments to total counterclockwise moments about

P

(300\text{ N} \times 1.0\text{ m}) + (200\text{ N} \times 2.0\text{ m}) = T \times 4.0\text{ m}

For rotational equilibrium, the sum of clockwise moments about any pivot must equal the sum of counterclockwise moments about that same pivot.

Calculate the value of the tension force

T

T = \frac{700}{4.0} = 175\text{ N}

Simplifying the moment equation gives the magnitude of the upward supporting force.

Anahtar Kavram

Principle of Moments and Rotational Equilibrium

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