Motion Of An Insect In The Rough Bowl

Motion of an Insect in the Rough Bowl

Edited By Vishal kumar | Updated on Jul 02, 2025 05:43 PM IST

Consider that you are observing a small insect that is attempting to get out of a shallow, rough bowl. The small insect crawls and, in the process, experiences the roughness of the surface, thus trying to climb out. Sometimes, it slips and goes backwards. Sometimes, it succeeds. This scenario is both interesting to observe and a great example of how motion and friction on various surfaces work. Its movements help you understand the different kinds of forces coming into play to affect motion on rugged surfaces.

In this article, we will cover the concept of Motion Of An Insect In The Rough Bowl. This concept falls under the broader category of law of motion which is a crucial chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of two questions have been asked on this concept. But no direct question in NEET from this concept.

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Motion Of An Insect In the Rough Bowl
Solved ExampleBased On Motion Of An Insect In The Rough Bowl
Summary

Motion of an Insect in the Rough Bowl

Motion Of An Insect In the Rough Bowl

As the insect crawls up, limiting friction force decreases and the component of weight along the surface (driving force) will decrease. Let's assume the insect can crawl up to height 'h' before it starts slipping. At that moment the frictional force will be limiting friction force as shown in the figure.

Let m=mass of the insect, r=radius of the bowl, μ= coefficient of friction for limiting condition at point A.

R=mgcos⁡θ… (i) Limiting friction −fl=μR=μmgcos⁡θfll=mgsin⁡θ… (ii)

From equation (i) and (ii)-
mgsin⁡θ=μmgcos⁡θ
tan⁡θ=μtan2⁡θ=μ2
(r2−y2)y2=μ2y=r1+μ2r−h=r1+μ2 h=r[1+11+μ2]

Solved ExampleBased On Motion Of An Insect In The Rough Bowl

Example 1: An insect crawls up a hemispherical surface very slowly (see the figure). The coefficient of friction between the insect and the surface is 1/3. If the line joining the centre of the hemispherical surface to the insect makes an angle θ with the vertical, the maximum possible value of θ is given by:

1) cot−1⁡3
2) tan−1⁡3
3) sec−1⁡3
4) cosec−13

Solution:

As the insect crawls up, limiting friction force decreases and the component of weight along the surface (driving force) will decrease. Let's assume the insect can crawl upto angle $\theta$ before it starts slipping. At that moment the frictional force will be limiting friction force as shown in the figure.

From fig,

R=mgcos⁡θ Limiting friction −fl=μR=μmgcos⁡θ… For equilibrium- fl=mgsin⁡θ… (2) From equation (1) and (2)- mgcos⁡θ=μmgcos⁡θcot⁡θ=1μ=3θ=cot−1⁡3

Hence, the answer is option (1).

Example 2: If the coefficient of friction between on insect and bowl is μ and the radius of the bowl is r. Find the maximum height to which the insect can crawl up in the bowl.

1) r1+μ2
(2) rμ1+μ2
3) r[1−11+μ2]
4) ru1−μ2

Solution:

Motion an insect in the rough bowl -

Till the component of its weight along with the bowl is balanced by limiting frictional force, then only the insect crawls up the bowl, up to a certain height h

R=mgcos⁡θ......(I)
and
Fl=mgsin⁡θ ...... (ii)
Dividing (ii) by (i)
tan⁡θ=F1R=μ[ using Fl=μR]∴r2−y2y=μ or y=r1+μ2

So h=r−y=r[1−11+μ2],

∴h=r[1−11+μ2]

where,

h= height up to which insect can climb
m= mass of insect
r= radius of the bowl
μ= coefficient of friction

Example 3: An insect is at the bottom of a hemisphere ditch of radius 1 m. It crawls up the ditch but starts slipping after it is at height ' h ' from the bottom. If the coefficient of friction b/w the ground and the insect is 0.75. Then h is : (g=10 m/s2)

1) 0.6 m

2) 0.45 m

3) 0.2 m

4) 0.8 m

Solution:

Till the component of its weight along with the bowl is balanced by limiting frictional force, then only the insect crawls up the bowl, up to a certain height h

R=mgcos⁡θ
(i) and
Fl=mgsin⁡θ
Dividing (ii) by (i)
tan⁡θ=FFR=μ[ using Fl=μR]∴r2−y2y=μ or y=r1+μ2

So h=r−y=r[1−11+μ2],

∴h=r[1−11+μ2]

h=r[1−11+μ2]=1[1−11+(34)2]=1(1−45)=15=0.2 m

Hence, the correct answer is option (2).

Example 4: A block of mass m is placed on a surface with a vertical cross-
section given by y=x36. If the coefficient of friction is 0.5 ,t he maximum height above the ground at which the block can be placed without slipping is :

1) 16m
2) 23m
3) 13m
4) 12m

Solution:

As we learnt in

mgSinθ=μmgCosθtan⁡θ=μ⇒>y=x36dydx=x22=12x=1y=16m

Hence, the answer is the option (1).

Summary

In Short, the motion of an insect in a rough bowl is influenced by factors such as friction, gravity, and the bowl's curvature. The insect's path is determined by the balance between its own movement and the forces acting on it, often resulting in complex, non-linear trajectories. Understanding these dynamics can provide insights into the insect's behaviour and the physical principles at play.

Frequently Asked Questions (FAQs)

1. What forces act on an insect moving inside a rough bowl?

The main forces acting on the insect are gravity pulling it downward, the normal force from the bowl's surface pushing upward, and friction between the insect and the bowl's surface. Depending on the insect's motion, there may also be centripetal force if it's moving in a circular path.

2. How does the roughness of the bowl affect the insect's motion?

The roughness of the bowl increases friction between the insect and the surface. This higher friction allows the insect to grip the surface better, enabling it to climb the sides of the bowl more easily and potentially preventing it from sliding down as quickly.

3. Can an insect stay stationary on the side of a rough bowl?

Yes, an insect can remain stationary on the side of a rough bowl if the static friction force is strong enough to counteract the component of gravity parallel to the surface. The rougher the surface, the easier it is for the insect to maintain its position.

4. What determines the maximum angle at which an insect can stay stationary on the bowl's surface?

The maximum angle depends on the coefficient of static friction between the insect's feet and the bowl's surface. The insect can stay stationary as long as the angle is less than or equal to the arctangent of this coefficient of friction.

5. How does the insect's mass affect its ability to climb the bowl?

The insect's mass doesn't directly affect its ability to climb, as both the gravitational force and the friction force are proportional to mass. However, larger insects might have different foot structures or adhesive abilities that could impact their climbing performance.

6. What path would an insect likely follow if released from rest at the top edge of a smooth bowl?

On a smooth bowl, the insect would likely slide down along a straight path following the steepest gradient (the path of least resistance) due to gravity, assuming no initial velocity and negligible air resistance.

7. How would the path differ in a rough bowl compared to a smooth one?

In a rough bowl, the insect's path would be more irregular due to friction. It might move in shorter, jerky motions or even be able to control its descent by gripping the surface. The overall path might be longer and less direct compared to a smooth bowl.

8. What is the role of normal force in the insect's motion on the bowl's surface?

The normal force is crucial as it acts perpendicular to the bowl's surface, counteracting the component of gravity perpendicular to the surface. It also determines the magnitude of the friction force, which is proportional to the normal force.

9. How does the curvature of the bowl affect the insect's motion?

The bowl's curvature changes the direction of the normal force and the component of gravity tangent to the surface as the insect moves. This varying force direction can cause the insect's path to curve, potentially leading to circular or spiral motions.

10. What would happen if an insect tried to climb a perfectly vertical wall of the bowl?

On a perfectly vertical wall, the insect would need to rely entirely on its ability to grip the surface, as there would be no component of the normal force to support its weight. Many insects can do this using specialized foot structures or adhesive secretions.

11. How does the concept of static friction apply to an insect at rest on the bowl's surface?

Static friction is the force that prevents the insect from sliding when at rest. It acts parallel to the surface and can vary in magnitude up to a maximum value, allowing the insect to stay stationary on inclined surfaces of the bowl.

12. What is the difference between static and kinetic friction for the insect's motion?

Static friction applies when the insect is at rest or trying to start moving, while kinetic friction acts when the insect is already in motion. Generally, the coefficient of static friction is larger than that of kinetic friction, making it easier for the insect to stay still than to keep moving.

13. How would adding a small amount of water to the bowl affect the insect's motion?

A small amount of water could create a thin film on the bowl's surface, potentially reducing friction. This could make it harder for the insect to grip the surface, causing it to slip more easily and potentially changing its path of motion.

14. What role does the insect's leg structure play in its movement on the rough bowl?

The insect's leg structure is crucial for its movement. Many insects have specialized structures like claws or adhesive pads that increase friction and allow them to grip rough surfaces more effectively, enabling them to climb steep inclines or even vertical surfaces.

15. How does the principle of action-reaction apply to the insect's motion in the bowl?

The principle of action-reaction (Newton's Third Law) is evident in the insect's interaction with the bowl. As the insect pushes against the bowl's surface to move, the bowl exerts an equal and opposite force back on the insect, allowing it to propel itself.

16. What would happen to the insect's motion if the bowl was suddenly rotated?

If the bowl was suddenly rotated, the insect would experience an apparent force (like the centrifugal force in a rotating reference frame). This could cause the insect to slide or roll, depending on its grip on the surface and the speed of rotation.

17. How does air resistance affect the insect's motion in the bowl?

For most insects, air resistance has a negligible effect on their motion within the bowl due to their small size and relatively low speeds. However, for very light insects or at higher speeds, air resistance could slightly affect their trajectory or terminal velocity if falling.

18. What would happen if the insect tried to jump while on the curved surface of the bowl?

If the insect jumped, it would initially follow a parabolic path in the air. However, due to the curved surface, it would likely land at a different angle and position than on a flat surface. The exact landing spot would depend on the jump's initial velocity and the bowl's curvature.

19. How does the distribution of the insect's mass affect its stability on the bowl's surface?

The distribution of the insect's mass affects its center of gravity. An insect with a lower center of gravity relative to its body size will generally be more stable on the curved surface, making it less likely to topple over on steeper sections of the bowl.

20. What would happen to the insect's motion if the bowl's roughness varied across its surface?

If the bowl's roughness varied, the insect's motion would change as it moved across different areas. It might move more easily on rougher patches and struggle or slip on smoother sections. This could result in a complex, unpredictable path of motion.

21. How does the concept of mechanical energy conservation apply to the insect's motion in the bowl?

As the insect moves in the bowl, its mechanical energy (sum of kinetic and potential energy) would be conserved in an ideal, frictionless scenario. In reality, friction causes some energy to be dissipated as heat, reducing the insect's mechanical energy as it moves.

22. What would be the effect of increasing the bowl's size on the insect's motion, assuming the same surface properties?

Increasing the bowl's size while maintaining the same surface properties would result in a less curved surface from the insect's perspective. This could make it easier for the insect to move horizontally but might increase the distance it needs to travel to reach the top of the bowl.

23. How does the angle of inclination at different points in the bowl affect the insect's required effort to move?

The angle of inclination affects the component of gravity parallel to the surface, which the insect must overcome to move upwards. Steeper angles require more effort from the insect to climb, while shallower angles are easier to traverse.

24. What would happen if the insect was moving in a circular path along the side of the bowl?

If the insect moves in a circular path, it experiences a centripetal force provided by friction. This force prevents the insect from moving in a straight line and maintains the circular motion. The insect must also overcome the component of gravity tangent to its path.

25. How does the concept of work apply to the insect climbing the bowl?

Work is done by the insect when it climbs the bowl, as it's applying a force over a distance against gravity. The amount of work done is equal to the increase in the insect's gravitational potential energy as it gains height in the bowl.

26. What would happen to the insect's motion if the bowl was accelerating horizontally?

If the bowl was accelerating horizontally, the insect would experience an apparent force opposite to the direction of acceleration (similar to what we feel in an accelerating car). This could cause the insect to slide or roll if the friction isn't sufficient to keep it in place.

27. How does the concept of impulse relate to an insect colliding with the bowl's surface?

Impulse, which is the change in momentum, applies when the insect collides with the bowl's surface. The force of impact over the brief collision time results in a change in the insect's velocity. A rougher surface might lead to a larger impulse due to increased friction during collision.

28. What would be the effect of adding a small amount of oil to the bowl's surface?

Adding oil would significantly reduce the coefficient of friction between the insect and the bowl. This would make it much harder for the insect to grip the surface, likely causing it to slip and slide more easily, especially on inclined portions of the bowl.

29. How does the insect's speed affect its ability to stay on a curved path in the bowl?

As the insect's speed increases, it requires a larger centripetal force to maintain a curved path. If the friction force isn't sufficient to provide this centripetal force, the insect will tend to move in a straighter path, potentially leaving contact with the bowl's surface.

30. What would happen if the bowl was placed in a reduced gravity environment?

In a reduced gravity environment, the normal force and friction would decrease proportionally to the reduction in gravity. This would make it easier for the insect to climb steep surfaces but might also reduce its ability to grip the surface, potentially leading to more floating or bouncing motions.

31. How does the principle of momentum conservation apply if the insect collides elastically with the bowl's surface?

In an elastic collision between the insect and the bowl, the total momentum of the system (insect + bowl) is conserved. However, since the bowl is much more massive, it would barely move, and the insect would essentially reverse its velocity component perpendicular to the surface.

32. What would be the effect of the bowl material on the insect's motion?

The bowl material affects the coefficient of friction between the insect and the surface. Materials like rough ceramic might provide better grip than smooth plastic. Some materials might also have electrostatic properties that could influence the insect's ability to adhere to the surface.

33. How does the concept of torque apply to an insect trying to climb a steep section of the bowl?

Torque comes into play as the insect tries to rotate its body to climb a steep section. The insect must generate enough torque with its legs to overcome the rotational effect of gravity, which tries to topple it backwards.

34. What would happen if the insect was moving with constant speed along the rim of the bowl?

If the insect moves with constant speed along the rim, it's undergoing uniform circular motion. The required centripetal force is provided by friction. The normal force and static friction must balance the insect's weight and provide the necessary centripetal force to maintain this motion.

35. How does the shape of the insect's body affect its motion on the curved surface of the bowl?

The shape of the insect's body affects its stability and motion on the curved surface. A flatter, wider insect might be more stable but could find it harder to maneuver in tightly curved sections. A more cylindrical insect might roll more easily but could also navigate curved surfaces more readily.

36. What would be the effect of vibrating the bowl on the insect's motion?

Vibrating the bowl would introduce additional forces on the insect. Depending on the frequency and amplitude of vibration, this could cause the insect to lose grip and slide, or in some cases, even bounce. It could also potentially help the insect overcome static friction to initiate motion.

37. How does the concept of centripetal acceleration apply to an insect moving in a circular path in the bowl?

An insect moving in a circular path experiences centripetal acceleration directed towards the center of the circle. This acceleration is provided by the component of the friction force pointing towards the center of the circular path. The magnitude of this acceleration is v²/r, where v is the velocity and r is the radius of the circular path.

38. What would happen if the insect was placed in a bowl that was rotating about its vertical axis?

In a rotating bowl, the insect would experience an apparent outward force (often called the centrifugal force in the rotating frame). This would make it harder for the insect to move towards the center of the bowl and easier to move towards the rim, potentially causing it to climb higher on the bowl's sides than it would in a stationary bowl.

39. How does the concept of pressure apply to the insect's feet contacting the bowl's surface?

Pressure is the force per unit area applied by the insect's feet on the bowl's surface. Insects often have specialized structures that allow them to increase this pressure at points of contact, enhancing their ability to grip the surface. The pressure can vary as the insect shifts its weight or changes its posture.

40. What would be the effect of electrically charging the bowl on the insect's motion?

Electrically charging the bowl could introduce electrostatic forces between the bowl and the insect. Depending on the charge of the insect, this could either attract or repel it from the surface, potentially making it easier or harder for the insect to grip the bowl. It could also affect the insect's path of motion.

41. How does the concept of mechanical advantage apply if the insect uses its legs as levers to climb the bowl?

The insect's legs act as levers, providing a mechanical advantage as it climbs. By adjusting the relative lengths of the effort arm (from joint to where force is applied) and load arm (from joint to where weight is supported), the insect can generate larger forces to overcome gravity and friction.

42. What would happen if the bowl was gradually tilted while the insect was moving on its surface?

As the bowl is tilted, the effective gravitational force on the insect changes relative to the bowl's surface. This could cause the insect to slide if the new angle exceeds its ability to grip. The insect might need to adjust its path or exert more effort to maintain its position or continue moving in the desired direction.

43. How does the concept of equilibrium apply to an insect at rest on the bowl's surface?

An insect at rest on the bowl's surface is in equilibrium, meaning the sum of all forces acting on it is zero. The normal force, friction, and gravity are balanced. For rotational equilibrium, the sum of all torques about any point must also be zero, which is achieved by the insect adjusting its posture and the points of contact of its legs.

44. What would be the effect of adding small obstacles or ridges to the bowl's surface?

Small obstacles or ridges would complicate the insect's path. They could provide additional points for gripping, potentially making it easier for the insect to climb. However, they could also act as barriers that the insect needs to navigate around or over, possibly leading to a more complex and energy-intensive path.

45. How does the concept of friction angle relate to the insect's ability to stay stationary on the bowl's surface?

The friction angle is the maximum angle of an inclined plane at which an object will remain stationary without sliding. For the insect, if the local slope of the bowl exceeds the friction angle (determined by the coefficient of static friction), the insect will begin to slide unless it can actively grip the surface.

46. What would happen if the insect was moving on the outside surface of an inverted bowl?

On the outside of an inverted bowl, the insect would need to overcome gravity more directly. It would rely heavily on its ability to grip the surface, using specialized foot structures or secretions. The risk of falling would be higher, and the insect might need to exert more energy to maintain its position, especially on the steeper sections.

47. How does the concept of terminal velocity apply if the insect falls inside a very tall bowl?

If the insect falls inside a very tall bowl, it would theoretically reach terminal velocity when the air resistance equals its weight. However, due to the bowl's curved surface, the insect is likely to make contact with the sides before reaching true terminal velocity, unless the bowl is extremely large relative to the insect.

48. What would be the effect of changing the insect's coefficient of restitution in collisions with the bowl?

The coefficient of restitution determines how elastic a collision is. A higher coefficient would result in more elastic collisions, causing the insect to bounce more when it impacts the bowl's surface. A lower coefficient would lead to more inelastic collisions, with less bouncing and more energy lost in each impact.