If the bathroom door has a hinge at and opens into the bedroom, will it hit the wardrobe? Are there any changes you would suggest if the door is made wider?

Question

Homi · Accepted Answer

Understand the Question

The bathroom door lies along the $y$ -axis between $\text{B}_1(0, 1.5)$ and $\text{B}_2(0, 4)$ .
The door is hinged at $\text{B}_1(0, 1.5)$ and swings into the bedroom (towards the positive $x$ -axis) with a radius equal to its width.
The closest edge of the wardrobe to the door is at $x = 3$ .
If the swing radius is less than $3\text{ units}$ , the door will clear the wardrobe without collision.

Step 1 · Calculate Door Width

From the plan, the coordinates of the door endpoints are $\text{B}_1 = (0, 1.5)$ and $\text{B}_2 = (0, 4)$ . Diagram 1

\begin{aligned} \text{Bathroom door width} &= y_{\text{B}_2} - y_{\text{B}_1} \\[0.6em] &= 4 - 1.5 \\[0.6em] &= 2.5 \text{ units} \end{aligned}

Step 2 · Check for Collision with the Wardrobe

The door is hinged at $\text{B}_1(0, 1.5)$ and swings into the bedroom, sweeping an arc with a radius of $2.5\text{ units}$ in the positive $x$ -direction.

The nearest boundary of the wardrobe starts at $\text{W}_1(3, 0)$ and $\text{W}_4(3, 2)$ at $x = 3$ .

\begin{aligned} \text{Door's maximum reach} &= 2.5 \text{ units} \\[0.6em] \text{Wardrobe's nearest } x\text{-position} &= 3 \text{ units} \end{aligned}

Since $2.5 < 3$ , the door will not hit the wardrobe.

Step 3 · Suggest Changes for a Wider Door

If the door width is increased to $3\text{ units}$ or more, the swing radius will reach or exceed the wardrobe at $x = 3$ , resulting in a collision.

Possible suggestions:

Make the door open inward into the bathroom instead of the bedroom.
Shift the wardrobe further to the right (greater $x$ -coordinate).
Restrict the door width so that it remains less than $3\text{ units}$ .

Answer

(i) No, the bathroom door will not hit the wardrobe as $2.5\text{ units} < 3\text{ units}$ .

(ii) If the door is made wider (width $\ge 3\text{ units}$ ), it will hit the wardrobe. Suggested changes include opening the door inwards into the bathroom, shifting the wardrobe to the right, or limiting the door width.

Common Mistakes

Measuring from Origin: Measuring the reach of the door from $(0, 0)$ instead of from the hinge location $\text{B}_1(0, 1.5)$ .
Ignoring the Swing Radius: Comparing only the $y$ -coordinates of the door and wardrobe rather than the maximum radial distance the door reaches along the $x$ -axis as it opens.

Question 2

More questions in Exercise 1.2