Assertion (A): In uniform circular motion, angular acceleration is zero.
Reason (R): In uniform circular motion, acceleration 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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Assertion (A) is true because angular speed \(omega\) is constant, thus \(alpha = domega/dt = 0\). Reason (R) is false; in uniform circular motion, the *magnitude* of acceleration is constant, but its *direction* continuously changes, so the acceleration vector is not constant.
Assertion (A): A cyclist is cycling on a rough horizontal circular track with increasing speed. Then the net frictional force on cycle is always directed towards centre of the circular track.
Reason (R): For a particle moving in a circle, component of its acceleration towards centre, that is, centripetal acceleration should exist (except when speed is zero instantaneously).
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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Assertion (A) is false. For increasing speed, friction must provide both a tangential force (for speed increase) and a centripetal force (for circular path). Thus, the net frictional force is not solely towards the center. Reason (R) is true, centripetal acceleration is necessary for circular motion whenever speed is non-zero. Since A is false and R is true, and this specific option is not available, we choose option (4) as the closest available if compelled.
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): 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): In uniform circular motion of a particle, sum of power delivered to it by all the forces acting on the particle is zero.
Reason (R): In uniform circular motion dot product of two perpendicular vectors, force and velocity is always zero.
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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In UCM, net force is perpendicular to velocity, so power \(P = \vec{F} \cdot \vec{v} = Fv \cos{90^{\circ}}\) is zero. Thus, (A) is true. Reason (R) correctly states that the dot product of perpendicular vectors is zero, which explains (A).