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PHXI05:LAWS OF MOTION

363197 A bead $A$ can slide freely along a smooth rod bent in the form of a half circle of Radius $R$ . The system is set in rotation with a constant angular velocity $ω$ about a vertical axis $O O$ ’. Find the angle $θ$ corresponding to steady position of the bead.

1

\cos^{- 1} (\frac{R ω^{2}}{g})

2

\cos^{- 1} (\frac{g}{R ω^{2}})

3

\sin^{- 1} (\frac{g}{R ω^{2}})

4

\sin^{- 1} (\frac{R ω^{2}}{g})

Explanation:

Net force acting along the tangent is
$m g \sin θ - m r ω^{2} \cos θ = 0$
or $m g \sin θ [1 - \frac{r ω^{2}}{g} \cos θ] = 0$
or $θ = \cos^{- 1} (\frac{g}{R ω^{2}})$
supporting img

PHXI05:LAWS OF MOTION

363198 A body of mass $200 g$ is tied to a spring of spring constant $12.5 N / m$ , while the other end of spring is fixed at a point $O$ . If the body moves about $O$ in a circular path on a smooth horizontal surface with constant angular speed. Then the ratio of extension in the spring to its natural length will be

1

1 : 2

2

2 : 3

3

1 : 1

4

2 : 5

Explanation:

Given that, natural length of spring $= L$
Extension in the spring $= x$
Spring constant $k = 12.5 N / m$ ,
angular speed $ω = 5 r a d / s$
Here centripetal force is provided by spring force which is $K X$ .
Centripetal force is,
$k x = m (L + x) ω^{2} (1)$
Using eqn. (1), [ with ]
$\Rightarrow 12.5 x = \frac{1}{5} (L + x) 25 \Rightarrow 1.5 x = L$
$\Rightarrow \frac{x}{L} = \frac{2}{3}$
So, correct option is (2).

JEE - 2023

PHXI05:LAWS OF MOTION

363199 A moving train takes a turn. Here, we observe that:

1 The radius of curvature of outer rail will be greater than that of the inner rail

2 The radius of curvature of the inner rail will be greater than that of outer rail

3 The radius of curvature of one of the rails will be greater

4 The radius of curvature of the outer and inner rail will be the same

PHXI05:LAWS OF MOTION

363200 Assertion :
A cyclist always bends inwards while negotiating a curve.
Reason :
By bending, cyclist lowers his centre of gravity

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

The cyclist actually leans inward at an angle $θ$ to generate the necessary force for circular motion. If $N$ represents the 'Normal reaction " force from the ground, it can be divided into two components: horizontal and vertical. The vertical component, $N \cos θ$ , counter balances the cyclist's weight (mg), while the horizontal component, $N \sin θ$ , supplies the required centripetal force for circular motion.
$\Rightarrow$ Assertion is correct
As his head is now lowered the position of centre of gravity shifts towards ground slightly.
Reason is correct but is not direct explanation for assertion. So correct option is (2)

PHXI05:LAWS OF MOTION

363197 A bead $A$ can slide freely along a smooth rod bent in the form of a half circle of Radius $R$ . The system is set in rotation with a constant angular velocity $ω$ about a vertical axis $O O$ ’. Find the angle $θ$ corresponding to steady position of the bead.

1

\cos^{- 1} (\frac{R ω^{2}}{g})

2

\cos^{- 1} (\frac{g}{R ω^{2}})

3

\sin^{- 1} (\frac{g}{R ω^{2}})

4

\sin^{- 1} (\frac{R ω^{2}}{g})

Explanation:

Net force acting along the tangent is
$m g \sin θ - m r ω^{2} \cos θ = 0$
or $m g \sin θ [1 - \frac{r ω^{2}}{g} \cos θ] = 0$
or $θ = \cos^{- 1} (\frac{g}{R ω^{2}})$
supporting img

PHXI05:LAWS OF MOTION

363198 A body of mass $200 g$ is tied to a spring of spring constant $12.5 N / m$ , while the other end of spring is fixed at a point $O$ . If the body moves about $O$ in a circular path on a smooth horizontal surface with constant angular speed. Then the ratio of extension in the spring to its natural length will be

1

1 : 2

2

2 : 3

3

1 : 1

4

2 : 5

Explanation:

Given that, natural length of spring $= L$
Extension in the spring $= x$
Spring constant $k = 12.5 N / m$ ,
angular speed $ω = 5 r a d / s$
Here centripetal force is provided by spring force which is $K X$ .
Centripetal force is,
$k x = m (L + x) ω^{2} (1)$
Using eqn. (1), [ with ]
$\Rightarrow 12.5 x = \frac{1}{5} (L + x) 25 \Rightarrow 1.5 x = L$
$\Rightarrow \frac{x}{L} = \frac{2}{3}$
So, correct option is (2).

JEE - 2023

PHXI05:LAWS OF MOTION

363199 A moving train takes a turn. Here, we observe that:

1 The radius of curvature of outer rail will be greater than that of the inner rail

2 The radius of curvature of the inner rail will be greater than that of outer rail

3 The radius of curvature of one of the rails will be greater

4 The radius of curvature of the outer and inner rail will be the same

PHXI05:LAWS OF MOTION

363200 Assertion :
A cyclist always bends inwards while negotiating a curve.
Reason :
By bending, cyclist lowers his centre of gravity

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

The cyclist actually leans inward at an angle $θ$ to generate the necessary force for circular motion. If $N$ represents the 'Normal reaction " force from the ground, it can be divided into two components: horizontal and vertical. The vertical component, $N \cos θ$ , counter balances the cyclist's weight (mg), while the horizontal component, $N \sin θ$ , supplies the required centripetal force for circular motion.
$\Rightarrow$ Assertion is correct
As his head is now lowered the position of centre of gravity shifts towards ground slightly.
Reason is correct but is not direct explanation for assertion. So correct option is (2)

PHXI05:LAWS OF MOTION

363197 A bead $A$ can slide freely along a smooth rod bent in the form of a half circle of Radius $R$ . The system is set in rotation with a constant angular velocity $ω$ about a vertical axis $O O$ ’. Find the angle $θ$ corresponding to steady position of the bead.

1

\cos^{- 1} (\frac{R ω^{2}}{g})

2

\cos^{- 1} (\frac{g}{R ω^{2}})

3

\sin^{- 1} (\frac{g}{R ω^{2}})

4

\sin^{- 1} (\frac{R ω^{2}}{g})

Explanation:

Net force acting along the tangent is
$m g \sin θ - m r ω^{2} \cos θ = 0$
or $m g \sin θ [1 - \frac{r ω^{2}}{g} \cos θ] = 0$
or $θ = \cos^{- 1} (\frac{g}{R ω^{2}})$
supporting img

PHXI05:LAWS OF MOTION

363198 A body of mass $200 g$ is tied to a spring of spring constant $12.5 N / m$ , while the other end of spring is fixed at a point $O$ . If the body moves about $O$ in a circular path on a smooth horizontal surface with constant angular speed. Then the ratio of extension in the spring to its natural length will be

1

1 : 2

2

2 : 3

3

1 : 1

4

2 : 5

Explanation:

Given that, natural length of spring $= L$
Extension in the spring $= x$
Spring constant $k = 12.5 N / m$ ,
angular speed $ω = 5 r a d / s$
Here centripetal force is provided by spring force which is $K X$ .
Centripetal force is,
$k x = m (L + x) ω^{2} (1)$
Using eqn. (1), [ with ]
$\Rightarrow 12.5 x = \frac{1}{5} (L + x) 25 \Rightarrow 1.5 x = L$
$\Rightarrow \frac{x}{L} = \frac{2}{3}$
So, correct option is (2).

JEE - 2023

PHXI05:LAWS OF MOTION

363199 A moving train takes a turn. Here, we observe that:

1 The radius of curvature of outer rail will be greater than that of the inner rail

2 The radius of curvature of the inner rail will be greater than that of outer rail

3 The radius of curvature of one of the rails will be greater

4 The radius of curvature of the outer and inner rail will be the same

PHXI05:LAWS OF MOTION

363200 Assertion :
A cyclist always bends inwards while negotiating a curve.
Reason :
By bending, cyclist lowers his centre of gravity

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

The cyclist actually leans inward at an angle $θ$ to generate the necessary force for circular motion. If $N$ represents the 'Normal reaction " force from the ground, it can be divided into two components: horizontal and vertical. The vertical component, $N \cos θ$ , counter balances the cyclist's weight (mg), while the horizontal component, $N \sin θ$ , supplies the required centripetal force for circular motion.
$\Rightarrow$ Assertion is correct
As his head is now lowered the position of centre of gravity shifts towards ground slightly.
Reason is correct but is not direct explanation for assertion. So correct option is (2)

PHXI05:LAWS OF MOTION

363197 A bead $A$ can slide freely along a smooth rod bent in the form of a half circle of Radius $R$ . The system is set in rotation with a constant angular velocity $ω$ about a vertical axis $O O$ ’. Find the angle $θ$ corresponding to steady position of the bead.

1

\cos^{- 1} (\frac{R ω^{2}}{g})

2

\cos^{- 1} (\frac{g}{R ω^{2}})

3

\sin^{- 1} (\frac{g}{R ω^{2}})

4

\sin^{- 1} (\frac{R ω^{2}}{g})

Explanation:

Net force acting along the tangent is
$m g \sin θ - m r ω^{2} \cos θ = 0$
or $m g \sin θ [1 - \frac{r ω^{2}}{g} \cos θ] = 0$
or $θ = \cos^{- 1} (\frac{g}{R ω^{2}})$
supporting img

PHXI05:LAWS OF MOTION

363198 A body of mass $200 g$ is tied to a spring of spring constant $12.5 N / m$ , while the other end of spring is fixed at a point $O$ . If the body moves about $O$ in a circular path on a smooth horizontal surface with constant angular speed. Then the ratio of extension in the spring to its natural length will be

1

1 : 2

2

2 : 3

3

1 : 1

4

2 : 5

Explanation:

Given that, natural length of spring $= L$
Extension in the spring $= x$
Spring constant $k = 12.5 N / m$ ,
angular speed $ω = 5 r a d / s$
Here centripetal force is provided by spring force which is $K X$ .
Centripetal force is,
$k x = m (L + x) ω^{2} (1)$
Using eqn. (1), [ with ]
$\Rightarrow 12.5 x = \frac{1}{5} (L + x) 25 \Rightarrow 1.5 x = L$
$\Rightarrow \frac{x}{L} = \frac{2}{3}$
So, correct option is (2).

JEE - 2023

PHXI05:LAWS OF MOTION

363199 A moving train takes a turn. Here, we observe that:

1 The radius of curvature of outer rail will be greater than that of the inner rail

2 The radius of curvature of the inner rail will be greater than that of outer rail

3 The radius of curvature of one of the rails will be greater

4 The radius of curvature of the outer and inner rail will be the same

PHXI05:LAWS OF MOTION

363200 Assertion :
A cyclist always bends inwards while negotiating a curve.
Reason :
By bending, cyclist lowers his centre of gravity

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

The cyclist actually leans inward at an angle $θ$ to generate the necessary force for circular motion. If $N$ represents the 'Normal reaction " force from the ground, it can be divided into two components: horizontal and vertical. The vertical component, $N \cos θ$ , counter balances the cyclist's weight (mg), while the horizontal component, $N \sin θ$ , supplies the required centripetal force for circular motion.
$\Rightarrow$ Assertion is correct
As his head is now lowered the position of centre of gravity shifts towards ground slightly.
Reason is correct but is not direct explanation for assertion. So correct option is (2)

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