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NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) PDF Download

CONSTRAINED MOTION 

 

1.1 String constraint : 

When the two object are connected through a string and if the string have the following properties :

  • The length of the string remains constant i.e., it is inextensible string
  • Always remains taut i.e., does not slacks.

Then the parameters of the motion of the objects along the length of the string have a definite relation between them.

 

Ist format : - (when string is fixed) 

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

The block B moves with velocity v. i.e. each particle of block B moves with velocity v.

If string remain attached to block B it is necessary that velocity of each particle of string is same = v (vs = v)

Now we can say that Block A also moves with velocity v.

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) : If pulley is fixed then the velocity of all the particles of string is same along the string.

Ex.1

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) 

 

Sol. In the above situation block B is moving with velocity v. Then speed of each point of the string is v along the string.

speed of the block A is also v

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

 

Ex.2 

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

 

Sol. Q Block A is moving with velocity 8 ms-1.

velocity of every point on the string must be 8m/s along the string.

The real velocity of B is vB. Then the string will not break only when the compoent of vB along string is 8 m/s.

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

 

Ex.3 Find out the velocity of block B in a pulley block system as shown in figure.

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

 

Sol. In a given pulley block system the velocity of all the particle of string is let us assume v then.

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

10 m/s is the real velocity of block A then its component along string is v.

⇒ 10 cos 53° = v ...(1)

If vis the real velocity of block B then it component along string is v then

vBcos37° = v ...(2)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

from (1) & (2) vB cos37° = 10 cos53°

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

Ex.4

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

What is the velocity of block A in the figure as shown above. 

 

Sol. The component of velocity of ring along string = velocity of A

= NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) = vA ⇒ vA = 10 m/s

 

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) : In the first format only two points of string are attached or touched to moving bodies.

IInd format (when pulley is also moving)

To understand this format we consider the following example in which pulley is moving with velocity vp and both block have velocity vA & vB respectively as shwon in figure.

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

If we observe the motion of A and B with respect to pulley. Then the pulley is at rest. Then from first format.

vAP = - vBP 

(-ve sign indicate the direction of each block is opposite with respect to Pulley)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) : - To solve the problem put the values of vA, vB, & vP with sign.

Ex.5

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) 

 

Sol.NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) 

Putting vp = 10 ms-1, vB = 0,

we get

vA = 20 ms-1 (upward direction)

 

Ex.6 

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

 

Sol. If we take upward direction as ve then

10 = NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

vB = 25 m/sec (in upward direction)

 

Ex.7

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) 

Find out the velocity of Block D 

 

Sol. From 2nd format of constrained motion

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

 

Ex.8

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) 

Find the velocity of point G. 

 

Sol. In string ABCD from first format of constrain

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

60 + 10 = vG

vG = 70 m/s­

 

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) : In IInd format three or four Points of the string is attached to the moving bodies.

 

III format : 

 

solving strategy : 

1. First choose the longest string in the given problem which contains the point of which velocity/acceleration to be find out.

2. Now mark a point on the string wherever it comes in contact or leaves the contact of real bodies.

3. If due to motion of a point, length of the part of a string with point is related, increases then its speed will be taken + ve otherwise -ve.

 

Ex.9 

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) 

 

Sol.

Step 1. We choose a longest string ABCDEFGHIJ in which we have to find out velocity of point J (vc)

Step 2. Mark all the point A, B ................

Step 3. Write equation

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

 

Ex.10 

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

 

Find out the velocity of block E as shown in figure. 

 

Sol. 

Step-1 We first choose the longest string in which point j (block E) lie. (abcdefghij)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

Step 2 : Now write equation according to the velocity of each point (either increase or decrease the length)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) :  In the following figure pulley is moving with velocity v at an angle q with the horizontal.

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

Only v cos q is responsible to increase or decrease the length AB and v sin q is responsible to either decrease or increase the length CD.

* Further solving strategy is same as 3rd format

 

Ex.11 Find out the relation between acceleration a and b as shown in following figure. 

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) 

 

Sol. 

Step 1. Mark the points on the string which is attached to the real object (e.f,g,h)

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

Step 2. Acceleration of each point which are responsible to effect the length of string

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

 

2. Wedge Constraint : 

Conditions :

(i) Contact must not be lost between two bodies.

(ii) Bodies are rigid.

The relative velocity / acceleration perpendicular to the contact surface of the two rigid object is always zero. Wedge constraint is applicable for each contact.

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

In other words, Components of velocity and acceleration perpendicular to the contact surface of the two objects is always equal if there is no deformation and they remain in contact.

 

Ex.12 Find the relation between velocity of rod and that of the wedge at any instant in the figure shown. 

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT) 

 

Sol. Using wedge constraint.

NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)


 

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FAQs on NLM - Constrained Motion; Chapter Notes; Class 11; Physics (IIT-JEE and AIPMT)

1. What is constrained motion in physics?
Ans. Constrained motion refers to the motion of an object that is restricted or limited by some external force or constraint. In physics, a constraint is defined as a restriction that limits the motion of an object by imposing certain conditions on its movement. Examples of constraints include friction, gravity, tension, and other forces that act on an object.
2. What are the types of constraints in physics?
Ans. In physics, there are two types of constraints that govern the motion of an object: holonomic and non-holonomic constraints. Holonomic constraints are those that can be expressed in terms of the coordinates of the object, while non-holonomic constraints cannot. Examples of holonomic constraints include a ball rolling down an inclined plane or a pendulum swinging back and forth. Examples of non-holonomic constraints include a car skidding on ice or a ball moving through a fluid.
3. How is constrained motion relevant to IIT-JEE and AIPMT exams?
Ans. Constrained motion is a fundamental concept in physics that is often tested on exams like IIT-JEE and AIPMT. Students are expected to understand the principles of constrained motion and be able to apply them to solve problems related to motion and forces. Questions related to constrained motion often appear in the mechanics section of these exams, and a strong understanding of the topic is essential for success.
4. What is the difference between holonomic and non-holonomic constraints?
Ans. The main difference between holonomic and non-holonomic constraints is that holonomic constraints can be expressed in terms of the coordinates of the object, while non-holonomic constraints cannot. Holonomic constraints restrict the motion of an object to a particular path or surface, while non-holonomic constraints restrict the motion of an object in a more general way that cannot be described by a set of equations.
5. What are some examples of holonomic and non-holonomic constraints?
Ans. Examples of holonomic constraints include a ball rolling down an inclined plane, a pendulum swinging back and forth, or a car driving on a circular track. Examples of non-holonomic constraints include a ball moving through a fluid, a car skidding on ice, or a robot arm moving in three-dimensional space. Holonomic constraints are generally simpler to analyze and model mathematically, while non-holonomic constraints require more complex mathematical techniques and may be difficult to solve analytically.
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