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Step-by-Step Solution
Step 1: Identify the forces acting on the insect
• The insect of mass $m$ experiences a gravitational force $mg$ vertically downward.
• The ditch is hemispherical of radius $R = 1\,\text{m}$. Let the angle of the radius (from the center of the hemisphere to the insect) with the vertical be $\alpha$.
• The normal force $N$ from the surface of the hemisphere acts radially outward (perpendicular to the surface).
• Frictional force $f$ acts along the tangential direction of the hemisphere’s surface, opposing the tendency of slipping.
Step 2: Resolve weight into tangential and normal components
• Normal component: $N = mg\,\cos(\alpha)$.
• Component of weight down the slope (tangential): $mg\,\sin(\alpha)$.
Step 3: Condition for impending slip
• The maximum frictional force that can act is $f_{\max} = \mu N = \mu \,mg\,\cos(\alpha)$, where $\mu = 0.75$ is the coefficient of friction.
• The insect begins to slip when the downslope component of gravity equals the maximum frictional force:
\[
mg\,\sin(\alpha) \;=\; \mu\,mg\,\cos(\alpha).
\]
• Simplify by cancelling $mg$ on both sides:
\[
\sin(\alpha) \;=\; \mu\,\cos(\alpha)
\quad \Longrightarrow \quad
\tan(\alpha) \;=\; \mu \;=\; 0.75.
\]
Step 4: Determine the angle α
\[
\alpha \;=\; \arctan(0.75).
\]
A well-known right triangle with sides 3, 4, 5 gives
\[
\sin(\alpha) \;=\; 0.6,
\quad
\cos(\alpha) \;=\; 0.8.
\]
Step 5: Find the height h from the bottom of the hemisphere
• Let the center of the hemisphere be at the origin $(0,0)$ in a coordinate system with positive $y$ upwards. The bottom of the hemisphere is at $y = -R = -1\,\text{m}$.
• If the insect is at angle $\alpha$ (measured from the bottom, i.e., from the vertical at the center pointing downward), its vertical coordinate is
\[
y = -\,R\,\cos(\alpha).
\]
However, the height $h$ from the bottom is
\[
h = \bigl(\text{insect's }y \bigr) \;-\; \bigl(\text{bottom's } y\bigr)
= \bigl(-\,\cos(\alpha)\bigr) \;-\; \bigl(-1\bigr)
= 1 \;-\; \cos(\alpha).
\]
• Substituting $\cos(\alpha) = 0.8$,
\[
h \;=\; 1 \;-\; 0.8 \;=\; 0.2\,\text{m}.
\]
Step 6: Final Answer
Hence, the insect can climb up to a height of
\[
\boxed{0.20\,\text{m}}
\]
from the bottom before it starts slipping.
