A block of mass m is placed on a surface with a vertical cross-section given by y=x^3/6. If the coefficient of friction is 0.5…

Q:A block of mass m is placed on a surface with a vertical cross-section given by y = x3/6. If the coefficient of friction is 0.5, the maximum height above the ground at which the block can be placed without slipping is

(a) 1/6m

(b)2/3m

(c)1/3m

(d) 1/2m

Ans: (a)

Sol:$\large y = \frac{x^3}{6}$

$\large \frac{dy}{dx}=\frac{x^2}{2}$

$\large tan\theta = \frac{x^2}{2}$

$\large 0.5 = \frac{x^2}{2}$

x2 = 1

$x = \pm 1$

y = 1/6

A block of mass m is at rest under the action of force F against a wall as shown in figure…

Q: A block of mass m is at rest under the action of force F against a wall as shown in figure. Which of the following statement is incorrect?

Numerical

(a)f=mg (where f is the frictional force)

(b)F=N (where N is the normal force)

(c)F will not produce torque

(d)N will not produce torque

Ans: (d)

Sol: For equilibrium ,

F = N

f = mg

and $\displaystyle \tau_N + \tau_f = 0$

Hence , τf ≠ 0

τN ≠ 0

What is the maximum value of the force F such that the block shows in the arrangement, does not move?

Q: What is the maximum value of the force F such that the block shows in the arrangement, does not move?

Numerical

(a)20 N

(b)10 N

(c)12 N

(d)15 N

Ans: (a)

Sol: For vertical equilibrium of block

$\large N = mg + F sin60$

$\large N = \sqrt{3}g + \frac{\sqrt{3}F}{2}$

For No motion

$\large f \ge F cos60$

$\large \mu N \ge F cos60$

$\large \frac{1}{2\sqrt{3}}(\sqrt{3}g + \frac{\sqrt{3}F}{2}) \ge \frac{F}{2}$

$\large g \ge F/2$

$\large F \le 2 g$

= 20 N

An insect crawl up a hemispherical surface very slowly (see the figure). The coefficient of friction between…

Q: An insect crawl up a hemispherical surface very slowly (see the figure). The coefficient of friction between the surface and the insect is 1/3. If the line joining the centre of the hemispherical surface to the insect makes an angle α with the vertical, the maximum possible value of α is given
Numerical

(a) cot α =3

(b) tan α =3

(c) sec α =3

(d) cosec α =3

Ans: (a)

A long horizontal rod has a bead which can slide along its length and is initially placed at a distance L from one end A of the rod…

Q: A long horizontal rod has a bead which can slide along its length and is initially placed at a distance L from one end A of the rod. The rod is set in angular motion about A with a constant angular acceleration α , if the coefficient if friction between the rod and bead is μ, and gravity is neglected, then the time after which the bead starts slipping is

(a)√(μ/α)

(b)μ/√α

(c)1/√μα

(d)Infinitesimal

Ans: (a)