Day: January 11, 2023

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202301111028 Solution to 2003-IBHL-I-4

Two forces of magnitudes 7\,\mathrm{N} and 5\,\mathrm{N} act at a point. Which one of the following is not a possible value for the magnitude of the resultant force?

A. 1\,\mathrm{N}
B. 3\,\mathrm{N}
C. 5\,\mathrm{N}
D. 7\,\mathrm{N}

Roughwork.

Suppose

\begin{aligned} \mathbf{F}_1 & = 7\sin\alpha\,\hat{\mathbf{i}} + 7\cos\alpha\,\hat{\mathbf{j}} \\ \mathbf{F}_2 & = 5\sin\beta\,\hat{\mathbf{i}} + 5\cos\beta\,\hat{\mathbf{j}} \\ \end{aligned}

where \alpha ,\beta\in [0,2\pi ).

\begin{aligned} \mathbf{F}_3 & = \mathbf{F}_1+\mathbf{F}_2 \\ & = (7\sin\alpha + 5\sin\beta )\,\hat{\mathbf{i}} + (7\cos\alpha +5\cos\beta )\,\hat{\mathbf{j}} \\ F_3 & = |\mathbf{F}_3| \\ & = \sqrt{(7\sin\alpha + 5\sin\beta )^2 + (7\cos\alpha +5\cos\beta )^2}\\ & = \sqrt{74+70(\sin\alpha\sin\beta +\cos\alpha\cos\beta )}\\ & = \sqrt{74+70\cos (\alpha-\beta)} \\ \end{aligned}

\begin{aligned} \because \quad & \begin{cases} \max (F_3) = \sqrt{74+70(1)} = 12 \\ \min (F_3) = \sqrt{74+70(-1)} = 2 \\ \end{cases} \\ \therefore\quad & F_3 \in [2,12] \\ \end{aligned}

One can actually do without vector analysis for the sake of a mental exercise.

This problem is not to be attempted.

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202301110923 Solution to 1980-AL-PHY-IA-26

A charged ball X is suspended by a string.

When a uniform electric field E is applied horizontally, the ball is displaced a horizontal distance a such that the string makes an angle \theta with the vertical.

Roughwork.

As always start with a free-body diagram. Hence, draw

Resolving components, write

\begin{cases} \textrm{Horizontal:}\quad & T\sin\theta = qE \\ \textrm{Vertical:}\quad & T\cos\theta = mg \\ \end{cases}

Squaring and summing, write

\begin{aligned} (T\sin\theta )^2+ (T\cos\theta)^2 &= (qE)^2 + (mg)^2\\ T & = \sqrt{q^2E^2+m^2g^2} \\ \end{aligned}

By considering the electric field strength E as an independent variable, and the angle of elevation \theta the dependent variable, make differentiation

\displaystyle{\mathrm{d}E = \bigg(\frac{T\cos\theta}{q}\bigg)\,\mathrm{d}\theta}\qquad(\theta\in [0,90^\circ])

This problem is not to be attempted.