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, velocity (vector) and acceleration (vector) are varying due to changing direction. So (A) is true. The magnitude of centripetal acceleration is \(a_c = v^2/r\). So (R) is true. However, (R) describes the magnitude, not the reason for vector variation. Thus, (R) is not the correct explanation of (A).
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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In uniform circular motion, angular speed is constant, making angular acceleration \(alpha = 0\). So (A) is true. However, linear acceleration (centripetal acceleration) continuously changes direction, so it is not constant. Thus (R) is false.
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): 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).
Assertion (A): A body having uniform speed in circular path has a variable acceleration.
Reason (R): Direction of acceleration is always away from the centre.
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, speed is constant but velocity direction changes, so acceleration exists (centripetal). Thus (A) is true. Acceleration is towards the center, not away. So (R) is false.
Assertion (A): In turning a vehicle safely with uniform speed in circular path friction is static in nature and towards centre.
Reason (R): In turning a vehicle in circular path with increasing speed friction is kinetic in nature and tangential in direction.
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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For safe turning at uniform speed, static friction provides the necessary centripetal force towards the center. So (A) is true. If speed increases, kinetic friction might act but it's not tangential; it opposes relative motion. So (R) is false.
Assertion (A): In uniform circular motion, magnitude of acceleration is \(\frac{V^2}{R}\) and direction is always towards the centre.
Reason (R): In uniform circular motion, acceleration is constant.
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
In UCM, centripetal acceleration is \(a = \frac{V^2}{R}\) towards the center. So (A) is true. Acceleration direction continuously changes, so it's not constant. So (R) is false.