Motion Of Two Bodies One Resting On The Other

Q: What is the difference between relative motion and absolute motion in this context?

Relative motion refers to the motion of the upper body with respect to the lower body, which may be at rest or in motion. Absolute motion refers to the motion of either body with respect to a fixed reference frame, such as the ground.

Q: How does the surface area of contact between the two bodies affect their motion?

Contrary to common misconception, the surface area of contact does not affect the friction force or the likelihood of sliding in ideal conditions. The friction force depends on the normal force and the coefficient of friction, not the area of contact.

Q: How does Newton's Third Law apply to the interaction between the two bodies?

Newton's Third Law states that for every action, there is an equal and opposite reaction. The normal force exerted by the lower body on the upper body is balanced by an equal and opposite force exerted by the upper body on the lower body.

Q: How does the texture of the surfaces in contact affect the motion?

The texture of the surfaces affects the coefficient of friction between them. Rougher surfaces generally have higher coefficients of friction, making it less likely for the upper body to slide relative to the lower body.

Q: How does the concept of work apply when the upper body moves with the lower body without sliding?

When the upper body moves with the lower body without sliding, no work is done by friction because there is no displacement between the two surfaces. However, work is done by the force moving the entire system.

Q: What is meant by "motion of two bodies, one resting on the other"?

This concept refers to a situation where one object is placed on top of another, and the system as a whole is in motion. The bottom object moves, while the top object may or may not move relative to it, depending on factors like friction and applied forces.

Q: How does friction affect the motion of the upper body relative to the lower body?

Friction plays a crucial role in determining whether the upper body moves with the lower body or slides relative to it. If the static friction force is greater than the force tending to make the upper body slide, it will move with the lower body. If not, the upper body will slide relative to the lower body.

Q: What is the difference between static and kinetic friction in this scenario?

Static friction acts when the upper body is not moving relative to the lower body, preventing it from sliding. Kinetic friction comes into play when the upper body is sliding relative to the lower body, opposing the relative motion.

Q: How does the mass of the upper body influence its motion relative to the lower body?

The mass of the upper body affects the normal force between the two bodies, which in turn influences the maximum static friction force. A heavier upper body increases the friction force, making it more likely to move with the lower body without sliding.

Q: How does the coefficient of friction affect the motion of the two bodies?

The coefficient of friction determines the strength of the frictional force between the two bodies. A higher coefficient of friction makes it more likely for the upper body to move with the lower body without sliding, while a lower coefficient increases the likelihood of sliding.

Motion Of Two Bodies One Resting On The Other

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

To understand the motion of two bodies one resting on the other it as simply as possible, imagine you have put a book on top of a skateboard and given it a little nudge. While that skateboard is moving, the book is moving with the skateboard. These simple physical actions are supposed to involve a very interesting concept of motion and friction, which is quite in use in everyday life. The interaction of the two bodies, one resting on the other, helps us understand the dynamics of how the forces work together to create the motion.

In this article, we shall find out what happens when two bodies interact where one is at rest over the other one. This concept falls under the broader category of Laws 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 one question has been asked on this concept. And for NEET no direct questions were asked from this concept.

This Story also Contains

Block on Block Problems
When Force F is Applied to the Lower Body
Solved Example Based On Motion Of Two Bodies One Resting On The Other
Summary

Motion Of Two Bodies One Resting On The Other

Block on Block Problems

Let us assume two blocks of masses m_1,and m₂ are kept on a smooth horizontal surface as shown in the figure. The coefficient of friction between the blocks is μ. As the block of mass m₁is being fulled by applying force F the system starts moving.

To solve problems based on block-on-block situations, we must begin by calculating the limiting friction at all the rough surfaces. These values will help us determine the possibilities of relative motion between bodies in contact. As the surface is smooth block m₁will begin to move for any value of F. However the friction between the blocks will try to stop any relative motion between the blocks. Therefore, the friction force between the blocks will take any suitable direction and magnitude such that both blocks move with the same acceleration (no relative motion). In this case, the friction force between the blocks will be of static type. However, if we increase the value F, the requirement of friction force for no relative motion will also increase. The value of friction force can not be greater than the limiting value of friction.

To solve such problems we will initially assume that friction has successfully ensured that there is no relative motion between the blocks an determine the value of frictional force for such conditions. If this value of frictional force is less than the limiting friction, we can say that our assumption was right and all the results thus determined are the correct results.

But, if the value of frictional force comes to be more than limiting friction assuming no relative motion, then we know our assumption was incorrect as the value of friction force can not be greater than limiting friction. In such a case, since we have established that there is a relative motion between the blocks we must apply kinetic friction in a suitable direction and determine different accelerations of the blocks.

Let us first assume that there is no relative motion between the blocks. Both the blocks are moving with common acceleration a_cand the friction force between the blocks is f.

F.B.D of both blocks m_1,and m₂together-

F=(m1+m2)acac=Fm1+m2

F.B.D of the block of mass m_2-

N=m2 g

Limiting friction-
fl=μN=μm2 g

From newton's law-
f=m2ac=m2 F m1+m2

Case-1
f≤fl or m2 F m1+m2≤μm2 g

If this is the case then our assumption is correct and the blocks will move together with acceleration a_cand the friction force between them is static and equal to f. Where-

ac=Fm1+m2f=m2 F m1+m2

Case-2
f>fl or m2 F m1+m2>μm2 g

If this is the case, then our assumption is wrong as static friction can not be greater than limiting friction. In this case the the friction force acting between the blocks will be kinetic friction and both blocks will move with different accelerations. Let us assume the accelerations of blocks m₁, and m₂be a_1,and a₂respectively.

Kinetic friction-
fk=μN=μm2 g

Along horizontal direction-
F−fk=m1a1a1=F−μm2 g m1fk=m2a2a2=μg

Now, coming to the second part, when force is applied to the lower body.

When Force F is Applied to the Lower Body

Let's discuss possible 4 situations under this case:-

When there is no friction:-

B will move with acceleration (F/M) while A will remain at rest (relative to ground) as there is no pulling force on A.

aB=FM and aA=0

As relative to B, A will move backwards with acceleration (F/M) and so will fall from it in time t,

t=2La=2MLF

If friction is present between A and B only and

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F' (Pseudo force on body A) < F_l (limiting friction between body A and B):-

Both the bodies will move together with common acceleration, which is given by a=Fm+M

And the value of F' will be

F′=ma=Fm+M=mFm+M and Fl=μsmg as F′<Fl⇒mFm+M<μsmg⇒F<μs(m+M))g

So both bodies will move together with acceleration,

aA=aB=Fm+M if F<μs(m+M))g

If friction is present between A and B only and F > F'_l(limiting friction between body A and B):-

Fl′=μs(m+M))g

Both the bodies will move with different accelerations. Here force of kinetic friction μkmg will oppose the motion of B while will cause the motion of A.

maA=μkmg i.e aA=ukg

similarly,

F−Fk=MaB i.e. aB=[F−μkmg]M

As both, bodies are moving in the same direction,

Acceleration of body A relative to B will be given by:-

a=a_A-a_B=-\left[\frac{F-\mu_k g(m+M)}{M}\right]

A negative sign implies that relative to B, A will move backwards and will fall it after time,

t=2La=2MLF−μkg(m+M)

If there is friction between B and floor and F>F_l(limiting friction between body B and surface):-

Here Fl′′=μs(m+M)g

The system will move only if F>F_l then replacing F by $F-F''_l$

The entire situation (3) will be valid. However if

F<Fl′′,

The system will not move. This means friction between B and the floor will be F while between A and B is zero.

Solved Example Based On Motion Of Two Bodies One Resting On The Other

Example 1: Find the maximum value of F (in N) such that both blocks move together, given coefficient of friction between two blocks is 0.2 and the lower block is placed on a smooth surface. Take (g=10m/s²)

1)15

2)20

3)30

4)10

Solution:

As we keep on increasing the value of F, the value of friction force between blocks will also increase so as to stop relative motion between the blocks. for a maximum value of F, the friction force will also be maximum (Limiting friction).

let the common acceleration of blocks be a_c,and limiting friction between the blocks be f_l.

F.B.D of both blocks combined-

F=15ac…(1)

F.B.D of 5 kg block:

N=50fll=μN=0.2×50=10 N F−fl=5acF=10+5ac…(2)

From equation (1) and (2)-

F=15 N

Hence, the correct answer is option (1).

Example 2: The arrangement of the block system is shown in the figure. mass of block A is mA=5 kg and mass of B is mB= 10 kg. A constant force F = 100 N is applied on upper block A . Friction between A and B is μ and b/w B and the ground surface is smooth, then find the displacement (in meters) of block B in t = 2 sec if the system starts from rest.

1)5

2)4

3)3

4)8

Solution:

So draw FBD

For block A,N=mg and fK=μN
⇒μmg=μMng=0.5×5×10=25NfK=mBa=μmAg⇒a=μmAmB

For block B,
=0.5×5×1010=2.5 m/s2s=μt+1/2at2⇒s=1/2×2.5×4=5 m

Example 3: A block of mass 5kg and negligible dimensions are kept at the right edge of a plank of length 50cm and mass 10kg, as shown in the figure. The coefficient of friction between the plank and block is 0.5 and the ground is smooth. If the plank is being pulled with a force of 85N, the time taken for the block to fall over the plank is-

(g=10m/s²)

1)1.25 sec

2)0.75 sec

3)1 sec

4)Never

Solution:

Masse of block₌5kg

Mass of the plank=10 kg

Coefficient of friction between blocks, μ=0.5.

Length of Plank l=50 cm=0.5m

Only friction force is responsible for the motion of the block of mass 5 kg.

First, we need to check whether the block and the plank are moving together or separately for the given force of 85 N. For that, we will first assume the block and the plank are moving together with common acceleration a_c. Then, we will determine the maximum possible acceleration a_max of the block (when limiting friction is acting on the block). If a_maxis less than a_c, then our assumption is right otherwise block and plank are moving separately.

F.B.D of block and plank combined-

Fnet =msys a85=15acac=5.67 m/s2

F.B.D of the block-

N=50fl=μN=25 Nfl=5amaxamax=5 m/s2

Since, amax<ac

Both block and plank are moving separately. Therefore, Friction acting between the blocks will be kinetic.

Kinetic friction-

fk=μN=25 N

Let the acceleration of the plank be a₁ and, the acceleration of the block be a_2.

F.B.D of the block and the plank-

For, 5 kg block-
fk=5a2a2=5 m/s2

For 10 kg plank-
85−fk=10a1a1=6 m/s2

In the frame of reference of the plank-
Acceleration of block with respect to plank-
a21=a2−a1=−1 m/s2

Displacement of block before jumping off the planks21=−0.5 m

Let the time taken be 't' sec.

From second equation of motion -

s21=u21t+12a12t2−0.5=−0.5t2t=1sec

Hence, the answer is option (3).

amax=μg=37×9.8 F=(M+m)amax=5amax=21 Newton

Summary

In this article, we shall find out what happens when two bodies interact where one is at rest over the other one. We shall first look into how the friction between the two surfaces affects the motion of the two. In addition, we shall discuss the forces acting on both bodies, which include the normal and frictional forces. Further, we look at conditions under which the upper body is bound to slide and the impact of different surfaces and forces in such an instance. At the conclusion of this article, you should clearly understand the governing principles regarding the motion of two bodies in contact.

Frequently Asked Questions (FAQs)

1. What is the difference between relative motion and absolute motion in this context?

Relative motion refers to the motion of the upper body with respect to the lower body, which may be at rest or in motion. Absolute motion refers to the motion of either body with respect to a fixed reference frame, such as the ground.

2. How does the surface area of contact between the two bodies affect their motion?

Contrary to common misconception, the surface area of contact does not affect the friction force or the likelihood of sliding in ideal conditions. The friction force depends on the normal force and the coefficient of friction, not the area of contact.

3. How does Newton's Third Law apply to the interaction between the two bodies?

Newton's Third Law states that for every action, there is an equal and opposite reaction. The normal force exerted by the lower body on the upper body is balanced by an equal and opposite force exerted by the upper body on the lower body.

4. How does the texture of the surfaces in contact affect the motion?

The texture of the surfaces affects the coefficient of friction between them. Rougher surfaces generally have higher coefficients of friction, making it less likely for the upper body to slide relative to the lower body.

5. How does the concept of work apply when the upper body moves with the lower body without sliding?

When the upper body moves with the lower body without sliding, no work is done by friction because there is no displacement between the two surfaces. However, work is done by the force moving the entire system.

6. What is meant by "motion of two bodies, one resting on the other"?

This concept refers to a situation where one object is placed on top of another, and the system as a whole is in motion. The bottom object moves, while the top object may or may not move relative to it, depending on factors like friction and applied forces.

7. How does friction affect the motion of the upper body relative to the lower body?

Friction plays a crucial role in determining whether the upper body moves with the lower body or slides relative to it. If the static friction force is greater than the force tending to make the upper body slide, it will move with the lower body. If not, the upper body will slide relative to the lower body.

8. What is the difference between static and kinetic friction in this scenario?

Static friction acts when the upper body is not moving relative to the lower body, preventing it from sliding. Kinetic friction comes into play when the upper body is sliding relative to the lower body, opposing the relative motion.

9. How does the mass of the upper body influence its motion relative to the lower body?

The mass of the upper body affects the normal force between the two bodies, which in turn influences the maximum static friction force. A heavier upper body increases the friction force, making it more likely to move with the lower body without sliding.

10. How does the coefficient of friction affect the motion of the two bodies?

The coefficient of friction determines the strength of the frictional force between the two bodies. A higher coefficient of friction makes it more likely for the upper body to move with the lower body without sliding, while a lower coefficient increases the likelihood of sliding.

11. What is the role of acceleration in determining whether the upper body slides?

The acceleration of the lower body creates an inertial force on the upper body. If this force exceeds the maximum static friction force, the upper body will slide relative to the lower body. Higher accelerations are more likely to cause sliding.

12. What is the concept of the limiting angle of friction?

The limiting angle of friction is the maximum angle of inclination at which the upper body will remain at rest relative to the lower body without sliding. It depends on the coefficient of static friction between the two surfaces.

13. How does the direction of motion of the lower body affect the behavior of the upper body?

The direction of motion influences the direction of the inertial force on the upper body. For horizontal motion, this force acts horizontally. For vertical motion, it can increase or decrease the normal force, affecting the friction and likelihood of sliding.

14. What is the significance of the normal force in this scenario?

The normal force is crucial as it determines the magnitude of the friction force between the two bodies. It is equal to the weight of the upper body when on a horizontal surface, but can change on inclined surfaces or during vertical accelerations.

15. How does the angle of inclination affect the motion of two bodies on an inclined plane?

As the angle of inclination increases, the component of the upper body's weight parallel to the surface increases, making it more likely to slide. The normal force also decreases, reducing the maximum friction force available to prevent sliding.

16. How does the initial velocity of the system affect the likelihood of sliding?

The initial velocity itself doesn't affect the likelihood of sliding. It's the change in velocity (acceleration) that matters. A sudden change in velocity is more likely to cause sliding than a gradual change.

17. What happens if the upper body is not uniformly dense?

Non-uniform density in the upper body can affect its center of mass and potentially its stability on the lower body. This could influence the likelihood of tipping or sliding in certain situations.

18. How does the elasticity of the bodies affect their motion?

The elasticity of the bodies affects how they respond to forces and impacts. More elastic bodies may bounce or deform temporarily upon sudden changes in motion, potentially altering the contact area and friction dynamics.

19. How does the concept of mechanical advantage apply to the motion of two bodies?

Mechanical advantage isn't directly applicable in most cases of two bodies moving together. However, it could be relevant if there's a mechanism (like an inclined plane) involved in the relative motion between the bodies.

20. What is the significance of the coefficient of restitution in collisions involving this two-body system?

The coefficient of restitution becomes important if the two-body system collides with another object. It determines how much kinetic energy is conserved in the collision, affecting the post-collision motion of both bodies.

21. What happens if the lower body moves in a circular path?

Circular motion introduces centripetal acceleration. The upper body experiences an apparent outward force (centrifugal force in the rotating frame), which may cause it to slide outward if friction is insufficient.

22. What is the significance of the normal reaction time in rapid changes of motion?

The normal reaction time, though usually negligible in macroscopic scenarios, becomes relevant in extremely rapid changes of motion. It represents the delay in the propagation of forces through the bodies, potentially affecting their relative motion.

23. What is the role of static indeterminacy in analyzing the forces between the two bodies?

Static indeterminacy can occur when there are more unknown forces than available equations to solve them. This might happen in complex geometries or when considering internal forces within the bodies.

24. What is the significance of the moment of inertia in rotational scenarios involving these two bodies?

The moment of inertia becomes important if either body rotates or if the system as a whole undergoes rotational motion. It affects the rotational dynamics and stability of the system.

25. How does the principle of conservation of angular momentum apply to this scenario?

Conservation of angular momentum applies if the system rotates or if there's a change in the rotational state of either body. It's particularly relevant in scenarios where the upper body might tip or rotate relative to the lower body.

26. How does the concept of damping apply to oscillations or vibrations in this two-body system?

Damping can occur due to friction or other dissipative forces, affecting any oscillations or vibrations in the system. It tends to reduce the amplitude of vibrations over time, stabilizing the relative position of the bodies.

27. What is the significance of the coefficient of rolling friction if one body is cylindrical?

If one body is cylindrical and can roll, the coefficient of rolling friction becomes relevant. Rolling friction is typically much lower than sliding friction, potentially altering the dynamics of the system significantly.

28. What is the effect of scaling on the behavior of this two-body system?

Scaling affects the relative importance of different forces. In very small systems, surface forces like adhesion become more significant relative to body forces like weight, potentially altering the motion dynamics dramatically.

29. How does the concept of stress and strain apply to deformations at the interface between the two bodies?

Stress and strain analysis becomes relevant when considering deformations at the contact interface. These deformations, though often small, can affect the contact area and friction characteristics, especially under high loads or during impacts.

30. What is the role of Newton's First Law in understanding the motion of the upper body?

Newton's First Law explains why the upper body tends to remain at rest relative to the lower body when the system starts moving. It will continue to move with the lower body unless an unbalanced force (like excessive acceleration) causes it to slide.

31. What is the concept of inertial force, and how does it apply to this scenario?

Inertial force, also known as pseudo-force, is the apparent force felt by the upper body due to the acceleration of the lower body. It's not a real force but appears in the frame of reference of the accelerating lower body, tending to cause relative motion.

32. What happens when the acceleration of the lower body exceeds a certain threshold?

When the acceleration exceeds a threshold determined by the coefficient of static friction and the normal force, the static friction force is overcome, and the upper body begins to slide relative to the lower body.

33. What is the difference in energy considerations when the upper body slides versus when it doesn't?

When the upper body doesn't slide, the system moves as a single unit, and energy is conserved within the system. When sliding occurs, some kinetic energy is converted to heat due to friction between the surfaces.

34. What is the role of momentum in the motion of two bodies, one resting on the other?

The total momentum of the system is conserved, regardless of whether the upper body slides or not. However, the distribution of momentum between the two bodies can change if sliding occurs.

35. How does the shape of the bodies affect their motion in this scenario?

The shape of the bodies can affect the stability of the system and the distribution of the normal force. For example, a flat surface provides more stability than a curved one, potentially reducing the likelihood of sliding or tipping.

36. What happens if the lower body suddenly stops moving?

If the lower body suddenly stops, the upper body tends to continue moving due to inertia. Whether it slides forward depends on the deceleration rate and the friction between the surfaces.

37. How does the concept of impulse apply when the motion of the lower body changes abruptly?

Impulse, which is the product of force and time, becomes significant during abrupt changes in motion. A large impulse over a short time can overcome static friction and cause the upper body to slide.

38. What is the significance of the center of mass in this two-body system?

The center of mass of the combined system is important for understanding its overall motion. The distribution of mass between the two bodies affects the system's response to forces and its stability.

39. How does air resistance affect the motion of two bodies, one resting on the other?

Air resistance primarily affects the motion of the entire system, potentially causing deceleration. It typically has little direct effect on the relative motion between the two bodies unless they have very different shapes or surface areas exposed to air flow.

40. How does the concept of torque apply to this scenario?

Torque becomes relevant if there's a possibility of the upper body rotating or tipping relative to the lower body. This could occur due to non-uniform acceleration or if the force is applied off-center.

41. What is the effect of adding a third body to the system?

Adding a third body complicates the system, introducing additional interfaces where friction acts. The motion of each body now depends on its interactions with both other bodies, potentially leading to more complex behavior.

42. What is the role of static equilibrium in this scenario?

Static equilibrium occurs when the upper body remains at rest relative to the lower body, despite the system's motion. This happens when the static friction force is sufficient to overcome any tendency for relative motion.

43. How does the presence of lubricants between the two bodies affect their motion?

Lubricants reduce the coefficient of friction between the bodies, making it more likely for the upper body to slide relative to the lower body. This can significantly alter the dynamics of the system under acceleration.

44. What is the effect of temperature on the motion of two bodies, one resting on the other?

Temperature can affect the coefficient of friction between the bodies, potentially changing their relative motion. Extreme temperatures might also cause thermal expansion or contraction, altering the contact dynamics.

45. How does the concept of pressure apply to the interface between the two bodies?

Pressure at the interface is the normal force divided by the area of contact. While it doesn't directly affect friction in ideal conditions, very high pressures in real-world scenarios might alter surface properties and friction characteristics.

46. How does the concept of work-energy theorem apply to this two-body system?

The work-energy theorem relates the work done on the system to its change in kinetic energy. It applies to the system as a whole and can help analyze energy transfers, especially when sliding occurs.

47. How does the concept of simple harmonic motion apply to vibrations in this two-body system?

Simple harmonic motion can occur if the upper body is displaced slightly and experiences a restoring force proportional to the displacement. This might happen due to small imperfections or elasticity in the surfaces.

48. How does the concept of virtual work apply to the analysis of this two-body system?

Virtual work principles can be used to analyze the system's equilibrium and stability, especially in cases where direct force analysis is complex. It involves considering hypothetical displacements and the work done by forces.

49. What is the effect of non-uniform acceleration on the motion of the two bodies?

Non-uniform acceleration can lead to varying inertial forces over time, potentially causing complex relative motion between the bodies. It may lead to periods of sliding alternating with periods of moving together.

50. How does the presence of adhesive forces between the surfaces affect the motion?

Adhesive forces can increase the effective friction between the surfaces, making it less likely for the upper body to slide. This can be particularly significant for very smooth surfaces or in the presence of liquids forming capillary bridges.