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DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

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Introductory Exercise 5.2

Q1. Three blocks of mass 1 kg, 4 kg and 2 kg are placed on a smooth horizontal plane as shown in figure. Find: 
(a) the acceleration of the system, 
(b) the normal force between 1 kg block and 4 kg block, 
(c) the net force on 2 kg block.

DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: (a) 10 ms-2 (b) 110 N (c) 20 N
Sol: Acceleration of system
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
= 10 m/s2
Let normal force between 1 kg block and 4 kg block = F1
∴ Net force on 1 kg block = 120 - N
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
or 10 = 120 - F1
i.e., F1 = 110 N
Net force on 2 kg block = 2*a
= 2*10
=20 N


Q2. Two blocks of mass 2 kg and 4 kg are released from rest over a smooth inclined plane of inclination 30° as shown in figure. What is the normal force between the two blocks?

DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: zero
Sol: As, 4 g sin 30° > 2 g sin 30°
The normal force between the two blocks will be zero.


Q3. What should be the acceleration ‘a’ of the box shown in figure so that the block of mass m exerts a force mg/4 on the floor of the box?
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: 3g/4
Sol: DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET∴ N = mg/4
As lift is moving downward with acceleration a , the pseudo force on A will be ma acting in the upward direction. For the block to be at rest w.r.t. lift.
N + ma = mg
or DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET


Q4. A plumb bob of mass 1 kg is hung from the ceiling of a train compartment. The train moves on an inclined plane with constant velocity. If the angle of incline is 30°. Find the angle made by the string with the normal to the ceiling. Also, find the tension in the string, (g = 10 m/s2)

Ans: 30°, ION
Sol: Angle made by the string with the normal to the ceiling = θ = 30°
As the train is moving with constant velocity no pseudo force will act on the plumb-bob.
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETTension in spring = mg
= 1*10
= 10 N


Q5. Repeat both parts of the above question, if the train moves with an acceleration a = g/2 up the plane.

Ans: DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Sol: Pseudo force (= ma) on plumb-bob will be as shown in figure
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETT cos φ = mg + ma cos (90° - θ)
i.e., T cos φ = mg + ma sin θ …(i)
and       T sin φ = ma cos θ
Squaring and adding Eqs. (i) and (ii),
T2 = m2 g2 + m2 a2 sin2 θ + 2m2 ag sin θ + m2 a2 cos2 θ …(iii)
T2 = m2 g2 + m2 a2 + m2 ag (∴ θ = 30°)
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
or
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Dividing Eq. (i) by Eq. (ii),
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
i.e.,  DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET


Q6. Two blocks of mass 1 kg and 2 kg are connected by a string AS of mass 1 kg. The blocks are placed on a smooth horizontal surface. Block of mass 1 kg is pulled by a horizontal force F of magnitude 8 N. Find the tension in the string at points A and B.
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: 4 N, 6 N
Sol: DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETDC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETNet force on 1 kg mass = 8 - T2
∴ 8 - T2 = 1*2
⇒ T2 = 6 N
Net force on 1 kg block = T1
∴ T1 = 2a = 2*2 = 4 N


Introductory Exercise 5.3

Q1. In the arrangement shown in figure what should be the mass of block A, so that the system remains at rest? Neglect friction and mass of strings.

Ans: 3 kg
Sol: F = 2 g sin 30° = g
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETFor the system to remain at rest
T2 = 2 g …(i)
T2 + F = T1 …(ii)
or          T2 + g = T1 …[ii (a)]
T1 = mg …(iii)
Substituting the values of T1 and T2 from Eqs. (iii) and (i) in Eq. [ii(a)]
2 g + g = mg
i.e., m = 3 kg


Q2. In the arrangement shown in figure, find the ratio of tensions in the strings attached with 4 kg block and that with 1 kg block.
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: 4
Sol: 
As net downward force on the system is zero, the system will be in equilibrium
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET∴  T1 = 4 g
and T2 = 1 g
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET


Q3. Two unequal masses of 1 kg and 2 kg are connected by a string going over a clamped light smooth pulley as shown in figure. The system is released from rest. The larger mass is stopped for a moment 1.0 s after the system is set in motion. Find the time elapsed before the string is tight again.
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: 1/3 s
Sol: 2 g - T = 2a
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETT - 1 g = 1a
Adding above two equations
1 g = 3 a
∴ a = g/3
Velocity of 1kg block 1 section after the system is set in motion
v = 0 + at
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
On stopping 2 kg, the block of 1kg will go upwards with retardation g. Time ( t' ) taken by the 1 k g block to attain zero velocity will be given by the equation.
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
⇒   DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
If the 2 kg block is stopped just for a moment (time being much-much less than 1/3 s), it will also start falling down when the stopping time ends.
In DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET time upward displacement of 1 kg block
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Downward displacement of 2 kg block
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
As the two are just equal, the string will again become taut after time 1/3 s.


Q4. Two unequal masses of 1 kg and 2 kg are connected by an inextensible light string passing over a smooth pulley as shown in figure. A force F = 20 N is applied on 1 kg block. Find the acceleration of either block. (g = 10 m/s2).
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Sol: F + 1g - T = 1a      … (i)
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETand T - 2g = 2a …(ii)
Adding Eqs. (i) and (ii),
F - 1g = 3a
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET


Introductory Exercise 5.4

Q1. Consider the situation shown in figure. Both the pulleys and the string are light and all the surfaces are smooth.
(a) Find the acceleration of 1 kg block.
(b) Find the tension in the string.
(g = 10 m/s 2).
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: (a) 2g/3, (b) 10/3 N
Sol: 2T = 2*a            … (i)
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETand     1g - T = 2a …(ii)
Solving Eqs. (i) and (ii),
α = g/3
∴  Acceleration of 1 kg block
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Tension in the string
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET


Q2. Calculate the acceleration of either blocks and tension in the string shown in figure. The pulley and the string are light and all surfaces are smooth.
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Sol: Mg - T = Ma             …(i)
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETT = Ma …(ii)
Solving Eqs. (i) and (ii)
α = g/2
and T = Mg


Q3. Find the mass M so that it remains at rest in the adjoining figure. Both the pulley and string are light and friction is absent everywhere, (g = 10 m/s 2).

Ans: 4 . 8 kg
Sol: Block of mass M will be at rest if
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETT = Mg …(i)
For the motion of block of mass 3 kg
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET  ....(ii)
For the motion of block of mass 2 kg
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET  ....(iii)
Adding Eqs. (ii) and (iii),
g = 5a
i.e.,  α = g/5
Substituting above value of a in Eq. (iii),
T/2 = 2(g+a)
or  DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
= 24g/5
Substituting value of above value of T in Eq. (i),
M = 24/5
= 4.8 kg


Q4. In figure assume that there is negligible friction between the blocks and table. Compute the tension in the cord connecting m2 and the pulley and acceleration of m2 if m1 = 300 g, m2 = 200g and F = 0.401V.
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ans: DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Sol: T/2 = m1.2a
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEETi.e., T = 4 m1a …(i)
F - T = m2a …(ii)
or      F - 4m1a = m2a
or   DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
∴ T = 4 m1a
= 4*0.3* 2/7
= 2.4/7
= 12/35 N



Introductory Exercise 5.5

Ques 1: In figure m1 = 1 kg and m2 = 4 kg. Find the mass M o f the hanging block which will prevent the smaller block from slipping over the triangular block. All the surfaces are friction less and the strings and the pulleys are light
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Note In exercises 2 to 4 the situations described take place in a box car which has initial velocity v = 0 but acceleration
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Ans: 6 .83 kg
Sol: Block on triangular block will not slip if
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
m1a cos θ= m1g sin θ
i.e., a = g tanθ …(i)
N = m1g cosθ + m1asinθ …(ii)
For the movement of triangular block
T - N sinθ = m2α …(iii)
For the movement of the block of mass M
Mg - T = Ma …(iv)
Adding Eqs. (iii) and (iv),
Mg - N sinθ = ( m2 + M )α
Substituting the value of N from Eq. (ii) in the above equation
Mg - (m1g cosθ + m1α sinθ) sinθ
=( m2α + Mα)
i.e., M (g - α) = m1g cosθ sinθ + (m2 + m1 sin2θ)α
Substituting value of a from Eq. (i) in the above equation,
M(1 - tanθ) = m1 cosθ sinθ + ( m2 + m1 sin2 θ) tanθ
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Substituting θ = 30° , m1 = 1 kg and m2 = 4 kg
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
= 6.82 kg

Ques 2: A 2 kg object is slid along the friction less floor with initial velocity (10 m/s)î (a) Describe the motion of the object relative to car (b) when does the object reach its original position relative to the box car.
Ans:
(a) x = x0 + 10t - 2.5t2 , v = 10 - 5t (b) t = 4s
Sol: (a) Using DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Displacement of block at time t relative to car would be
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Velocity of block at time t (relative to car) will be
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
(b) Time (t) for the block to arrive at the original position (i.e., x = x0) relative to car
x0 = x+ 10 t - 2.5 t2
⇒ t = 4s

Ques 3: A 2 kg object is slid along the friction less floor with initial transverse velocity (10 m/s)DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET. Describe the motion (a) in car’s frame (b) in ground frame.
Ans: 
(a) x = x0 - 2.5t2, z = z0 + 10t, vx = - 5t, vz = 10 ms-1 
(b) x = x0, z = z0 + 10t, vx = 0, vz = 10 ms-1
Sol: (a) In car’s frame position of object at time t would be given by      
In car’s frame
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
x = x0 + 0*t + 1/2(-5)t2
i.e., x = x0 - 2.5 t2 …(i)
and z = z + 10t …(ii)
Velocity of the object at time t would be
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
and
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
(b) In ground frame the position of the object at time t would be given by
In ground frame
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
x = x0
and z = z0 + 10t
Velocity of the object at time t would be
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
ans
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

Ques 4: A 2 kg object is slid along a rough floor (coefficient of sliding friction = 0.3) with initial velocity (10 m/s)î. Describe the motion of the object relative to car assuming that the coefficient of static friction is greater than 0.5.
Ans: 
x = x0 + 10f - 4t2 , K = 10 - 8 t for 0<f<1.25 s object stops at t = 1.25 s and remains at rest relative to car.
Sol: m = 2 kg
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Normal force on object = mg
Maximum sliding friction = μsmg
= 0.3*2*10 = 6 N
Deceleration due to friction = 6/2 =3 m/s2
Deceleration due to pseudo force  = 5 m/s2
∴ Net deceleration = (3 + 5) m/s2
= 8 m/s2
∴ Displacement of object at any time t (relative to car)
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Thus, velocity of object at any time t (relative to car)
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
The object will stop moving relative to car when
10 - 8t = 0 i.e., t = 1.25s
∴ vx = 10 - 8t for  0<t<1.25 s

Ques 5: A block is placed on an inclined plane as shown in figure. What must be the frictional force between block and incline if the block is not to slide along the incline when the incline is accelerating to the right at DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
Ans: 
9/25 mg
Sol: For block not to slide the frictional force ( f ) would be given by
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
f + ma cosθ = mg sinθ
or
f = mg sinθ - ma cosθ
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET
DC Pandey Solutions: Laws of Motion - 2 - Notes | Study Physics Class 11 - NEET

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