Assertion (A): If a body is in state of uniform circular motion then its velocity and acceleration both are varying.
Reason (R): If magnitude of velocity is \(v\) and radius of uniform circular motion is \(r\) then magnitude of acceleration is \(v^2/r\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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In uniform circular motion, speed is constant, but velocity (direction) and acceleration (direction) vary, making (A) true. Reason (R) gives the correct magnitude of centripetal acceleration \(a = v^2/r\), so (R) is true. However, (R) describes the magnitude, not why the vectors are varying, so it's not the correct explanation.
Assertion (A): In circular motion acceleration is always towards centre.
Reason (R): In uniform circular motion velocity is constant.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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In circular motion, centripetal acceleration is always directed towards the center. So (A) is true. In uniform circular motion, the speed is constant, but the direction of velocity changes continuously, meaning velocity is not constant. Hence (R) is false.
Assertion (A): If a particle is moving on a curved path its \( \frac{d|\vec{v}|}{dt} \) may be zero.
Reason (R): A particle can move on curved path without any acceleration.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For motion on a curved path, if the speed is constant, then \( \frac{d|\vec{v}|}{dt} = 0 \). So (A) is true. Curved path motion always requires a centripetal acceleration. Hence (R) is false.
Assertion (A): A cyclist must adopt a zig-zag path while ascending a steep hill.
Reason (R): The zig-zag path prevent the cyclist to slip down.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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When a cyclist takes a zig-zag path, the effective angle of inclination \( \alpha \) becomes smaller than the actual angle \( \theta \). This reduces the component of gravity along the slope \( mg \sin \alpha \) and increases the normal force \( mg \cos \alpha \). This makes it easier to ascend and helps prevent slipping. Both (A) and (R) are true and (R) is the correct explanation of (A).
Assertion (A): Infinitesimally small angular displacement is a vector quantity.
Reason (R): Angular velocity doesn’t depend upon reference frame.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Infinitesimally small angular displacement \( d\vec{\theta} \) is a vector because it obeys the commutative law of vector addition. Thus (A) is true.
Angular velocity \( \vec{\omega} \) is a vector quantity, and its value depends on the chosen reference frame. Hence (R) is false.
Assertion (A): A bob of mass \( m \) is freely suspended from a light rod of length \( L \). The minimum speed given to bob at lowest position to complete vertical circle is \( 2\sqrt{gL} \).
Reason (R): A bob of mass \( m \) is freely suspended from a light string of length \( L \). If bob is given speed \( \sqrt{6gL} \) at the lower position then bob will be complete vertical circle.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For a mass attached to a rod, the minimum speed at the lowest point to complete a vertical circle is \( v_{min} = 2\sqrt{gL} \). So (A) is true. For a mass on a string, if \( v_{bottom} = \sqrt{6gL} \), the speed at the top will be \( v_{top} = \sqrt{2gL} \). Since \( v_{top} > \sqrt{gL} \), the circle will be completed. So (R) is true. However, they describe different conditions, so (R) is not a correct explanation of (A).
Assertion (A): Average angular velocity is a scalar quantity.
Reason (R): Large angular displacements \( (\Delta \theta) \) is a scalar.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Instantaneous angular velocity \( \vec{\omega} \) is a vector. However, finite angular displacement \( \Delta \theta \) is not a vector, but a scalar, as stated in Reason (R). Therefore, if average angular velocity is defined as the scalar \( \Delta \theta / \Delta t \), then Assertion (A) is considered true. In this context, both (A) and (R) are true, and (R) provides the explanation for (A).
Assertion (A): During a safe turn, with constant speed the value of centripetal force should be less than or equal to the limiting frictional force.
Reason (R): The centripetal force is provided by the frictional force between the tyre and the road.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For a vehicle to take a safe turn on a flat road, the required centripetal force \( (mv^2/r) \) must be provided by the static frictional force between the tires and the road. This frictional force has a limiting maximum value \( f_{s,max} = \mu_s N \). Therefore, for a safe turn, the centripetal force must be less than or equal to this limiting frictional force.
Both (A) and (R) are true, and (R) correctly explains (A).
Assertion (A): If the speed of a body is constant, the body cannot have a path other than a circular or straight line path.
Reason (R): It is not possible for a body to have a constant speed in an accelerated motion.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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A body can have constant speed and follow a curvilinear path (e.g., parabolic trajectory if air resistance is ignored). So (A) is false. A body can have constant speed but changing direction, leading to acceleration (e.g., UCM). So (R) is false.
Assertion (A): In circular motion, centripetal and centrifugal forces act in opposite directions and balance each other.
Reason (R): Centripetal force is a pseudo force.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Centripetal force is a real force causing circular motion. Centrifugal force is a pseudo force in a non-inertial frame. They are not interaction pairs and do not balance each other. So (A) is false. Centripetal force is a real force. So (R) is false.