A sphere of mass 2000 g and radius 5 cm is rotating at the rate of 300 rpm .Then the torque required to stop it in 2 π revolutions, is

1.6 × 10² dyne cm
1.6 × 10³ dyne cm
2.5 × 10⁴ dyne cm
2.5 × 10⁵ dyne cm

Solution

Auniform rod of length l is free to rotate in avertical plane about a fixed horizontal axis through O. The rod begins rotating from rest from its unstable equilibrium position. When it has turned through an angle θ, its angular velocity ω is given as

\(\sqrt{\frac{6g}{\iota }}sin\, \theta\)
\(\sqrt{\frac{6g}{\iota }}sin\, \frac{\theta }{2}\)
\(\sqrt{\frac{6g}{\iota }}cos\, \frac{\theta }{2}\)
\(\sqrt{\frac{6g}{\iota }}cos\, \theta\)

Solution

Acertain bicycle can go up a gentle incline with constant speed when the frictional force of ground pushing the rear wheel is F₂ = 4N. With what force F₁ must the chain pull on the sprocket wheel if R₁=5 cm and R₂ = 30 cm?

4 N
24 N
140 N
³⁵⁄₄ N

Solution

For no angular acceleration τ_net = 0

⇒ F₁ × 5= F₂ × 30 (given F₂ = 4N) ⇒ F₁ = 24 N

A wheel is rolling straight on ground without slipping. If the axis of the wheel has speed v, the instantenous velocity of a point P on the rim, defined by angle θ, relative to the ground will be

v cos(¹⁄₂ θ)
2v cos(¹⁄₂ θ)
v (1 + sin θ)
v (1 + cos θ)

Solution

Two flywheels A and B are mounted side by side with frictionless bearings on a common shaft. Their moments of inertia about the shaft are 5.0 kg m² and 20.0 kg m² respectively. Wheel A is made to rotate at 10 revolution per second. Wheel B, initially stationary, is now coupled to A with the help of a clutch. The rotation speed of the wheels will become

25 rps
0.5 rps
2 rps
None of these

Solution

By conservation of angular momentum,

5 × 10 = 5ω + 20ω ⇒ ω = ⁵⁰⁄₂₅ = 2 rps

A coin placed on a gramophone record rotating at 33 rpm flies off the record, if it is placed at a distance of more than 16cm from the axis of rotation. If the record is revolving at 66 rpm, the coin will fly off if it is placed at a distance not less than

1 cm
2 cm
3 cm
4 cm

Solution

When a bucket containing water is rotated fast in a vertical circle of radius R, the water in the bucket doesn’t spill provided

the bucket is whirled with a maximum speed of \(\sqrt{2gR}\)
the bucket is whirled around with a minimum speed of \(\sqrt{(1/2)gR}\)
the bucket is having arpm of \(\sqrt{900g/(\pi ^{2}R)}\)
the bucket is having a rpm of \(\sqrt{3600g/(\pi ^{2}R)}\)

Solution

A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is K. If radius of the ball be R,then the fraction of total energy associated with its rotational energy will be

\(\frac{R^{2}}{K^{2}+R^{2}}\)
\(\frac{K^{2}+R^{2}}{R^{2}}\)
^K²⁄_R²
\(\frac{K^{2}}{K^{2}+R^{2}}\)

Solution

A composite disc is to be made using equal masses of aluminium and iron so that it has as high a moment of inertiaas possible. This is possible when

the surfaces of the disc are made of iron with aluminium inside
the whole of aluminium is kept in the core and the iron at the outer rim of the disc
the whole of the iron is kept in the core and the aluminium at the outer rim of the disc
the whole disc is made with thin alternate sheets of iron and aluminium

Solution

Density of iron > density of aluminium moment of inertia = ∫r² dm.\

∴ Since ρ_iron > ρ_aluminium

so whole of aluminium is kept in the core and the iron at the outer rim of the disc.

If the linear density (mass per unit length) of a rod of length 3m is proportional to x, where x is the distance from one end of the rod, the distance of the centre of gravity of the rod from this end is

2.5 m
1 m
1.5 m
2 m

Solution

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