Hanging ball!

Classical Mechanics Level 2

A metal sphere is hung by a string fixed to a wall. The forces acting on the sphere is shown in the free body diagram below. Which of the following statement is incorrect?

Here T is the tension in the string, R is the normal reaction by the wall, and W is the weight of the ball.

R ⃗ + T ⃗ + W ⃗ = 0 \vec R + \vec T + \vec W = 0 T 2 = R 2 + W 2 {T^2} = {R^2} + {W^2} R = W tan ⁡ θ R = W\tan \theta T = R + W T = R + W

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Rohit Gupta by Rohit Gupta , 34, India

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2 solutions

Rohit Gupta
Aug 2, 2016

The three forces shown in the diagram are in equilibrium. Thus, their resultant is zero.
Therefore, the following equation will hold true,
T ⃗ + W ⃗ + R ⃗ = 0 \vec T + \vec W + \vec R = 0 Also, The tension force T is balancing the other two forces W and R. Therefore, T must have the same magnitude as the resultant of W and R. W and R are perpendicular to each other and the magnitude of their resultant is R 2 + W 2 \sqrt {{R^2} + {W^2}} . Hence, the following equation is also true,
T = R 2 + W 2 T = \sqrt {{R^2} + {W^2}}

The validity of the equation R = W tan ⁡ θ R = W\tan \theta can be established by breaking the tension T into its components along W and R. One of the component balances W and the other balances R. T cos ⁡ θ = W T sin ⁡ θ = R \begin{gathered} T\cos \theta = W \\ T\sin \theta = R \\ \end{gathered} Upon solving the above equations it can be concluded that R = W tan ⁡ θ R = W\tan \theta .

The equation left T = R + W T = R + W is incorrect, can be established by the mere fact that the equation T = R 2 + W 2 T = \sqrt {{R^2} + {W^2}} is true.

15 Helpful 4 Interesting 2 Brilliant 1 Confused

t=w+r r8 ans?

Atique Bin Siddique - 3 years, 5 months ago
Alekhya China
Jul 8, 2016

the forces R and W are together opposing the force T. want explanation?

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