A simple pendulum having a bob of mass m is suspended from the ceiling of a car used in a stunt film shooting. The car moves up along an inclined cliff at a speed v and makes a jump to leave the cliff and lands at some distance. Let R be the maximum height of the car from the top of the cliff. The tension in the string when the car is in air is.

1. mg

2. \(\mathrm{mg}-\frac{\mathrm{mv}^{2}}{\mathrm{R}}\)

3. \(m g+\frac{m v^{2}}{R}\)

4. zero

Let θ denote the angular displacement of a simple pendulum oscillating in a vertical plane. If the mass of the bob is m, the tension in the string is mg cosθ.

1. always

2. never

3. at the extreme positions

4. at the mean position

An object follows a curved (circular) path. The following quantities may remain constant during the motion.

Choose the correct option:

Assume that the earth goes round the sun in a circular orbit with a constant speed of 30 km/s.

1. The average velocity of the earth from 1st Jan, 90 to 30th June, 90 is zero

2. The average acceleration during the above period is 60 km/s²

3. The average speed from 1st Jan, 90 to 31st Dec, 90 is zero

4. The instantaneous acceleration of the earth points towards the sun

The position vector of a particle in a circular motion about the origin sweeps out equal area in equal time. Its

(a) velocity remains constant

(b) speed remains constant

(c) acceleration remains constant

(d) tangential acceleration remains constant

Choose the correct option:

1. (a) and (d)

2. (b) and (d)

3. (c) and (d)

4. (a) and (c)

A particle is going in a spiral path as shown in the figure with constant speed.

A car of mass M is moving on a horizontal circular path of radius r. At an instant its speed is v and is increasing at a rate a.

(a) The acceleration of the car is towards the centre of the path

(b) The magnitude of the frictional force on the car is greater than \(\frac{mv^2}{r}\)

(c) The friction coefficient between the ground and the car is not less than a/g

(d) The friction coefficient between the ground and the car is µ = tan^–1 \(\frac{v^2}{rg}\)

Choose the correct option:

1. (a) and (b)

2. (b) and (c)

3. (c) and (d)

4. All of these

A circular road of radius r is banked for a speed v = 40 km/hr. A car of mass m attempts to go on the circular road. The friction coefficient between the tyre and the road is negligible.

(a) The car cannot make a turn without skidding

(b) If the car turns at a speed less than 40 km/hr, it will slip down

(c) If the car turns at the correct speed of 40 km/hr, the force by the road on the car is equal to \(\frac{mv^2}{r}\)

(d) If the car turns at the correct speed of 40 km/hr, the force by the road on the car is greater than mg as well as greater than \(\frac{mv^2}{r}\)

Choose the correct option:

1. (b) and (d)

2. (c) and (d)

3. (a) and (d)

4. (a), (b) and (c)

A person applies a constant force \(\overrightarrow F\) on a particle of mass m and finds that the particle moves in a circle of radius r with a uniform speed v as seen from an inertial frame of reference.

(a) This is not possible

(b) There are other forces on the particle

(c) The resultant of the other forces is \(\frac{mv^2}{r}\) towards the centre

(d) The resultant of the other forces varies in magnitude as well as in direction

Choose the correct option:

1. (a) and (b)

2. (b) and (c)

3. (c) and (d)

4. (b) and (d)

Consider the following two statements