In the arrangement shown in the figure, a weight A , possesses mass m . A pulley B , possesses mass M . Also known are the moment of inertia I , of the pulley relative to its axis and the radii of the pulley R and 2 R . The mass of the threads is negligible. Find the acceleration of the weight A , after the system is set free.

Let T be the tension, ω be the angular acceleration, I be the moment of inertia. The forces acting on the pulley and the weight A are, as shown in the figure. Writing the force F = m ω and torque N = I β equations for the pulley, we get, M g + T A - 2 T = M ω B . . . ( 1 ) and, 2 T R + T A ( 2 R ) = I β = I ω B R . . . ( 2 ) Similarly, the force equation F = m ω , for the weight A , m g - T A = m ω A . . . ( 3 ) Since the threads are not slipping on the pulley, we can write the relation between accelerations of A a n d B as, ω A = ω B + 2 β R = ω B + 2 ω B = 3 ω B . . . ( 4 ) Now, removing T from equations 1 and 2 , we get, M g + 3 T A = M + I R 2 ω B , but T A = m g - m ω A and ω B = ω A 3 , so we get, M g + 3 m g - m ω A = M + I R 2 ω A 3 . On solving , we get, ω A = 3 ( M + 3 ω ) g M + 9 m + I R 2

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The arrangement shown in figure consists of two identical uniform solid cylinders, each of mass $m$ , on which two light threads are wound symmetrically. Find the tension of each thread in the process of motion. The friction in the axle of the upper cylinder is assumed to be absent.

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>DYNAMIC OF A SOLID BODY>Q 1.259.

In the arrangement shown in the figure, a weight $A$ , possesses mass $m$ . A pulley $B$ , possesses mass $M$ . Also known are the moment of inertia $(1), (2)$ , of the pulley relative to its axis and the radii of the pulley $R$ and $2 R$ . The mass of the threads is negligible. Find the acceleration of the weight $A$ , after the system is set free.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>DYNAMIC OF A SOLID BODY>Q 1.260.

A uniform solid cylinder $A$ , of mass $m_{1}$ , can freely rotate about a horizontal axis fixed to a mount $B$ , of mass $m_{2}$ . A constant horizontal force $F$ , is applied to the end $K$ , of a light thread tightly wound on the cylinder. The friction between the mount and the supporting horizontal plane is assumed to be absent. Find:

$(a)$ the acceleration of the point $K$
$(b)$ the kinetic energy of this system, $t seconds$ after the beginning of motion.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>DYNAMIC OF A SOLID BODY>Q 1.261.

A plank of mass $m_{1}$ , with a uniform sphere of mass $m_{2}$ placed on it, rests on a smooth horizontal plane. A constant horizontal force $F$ is applied to the plank. With what accelerations will the plank and the centre of the sphere move, provide there is no sliding between the plank and the sphere?

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>DYNAMIC OF A SOLID BODY>Q 1.262.

A uniform solid cylinder of mass

m

and radius

R

, is set in rotation, about its axis, with an angular velocity

ω_{0},

then lowered with its lateral surface onto a horizontal plane and released. The coefficient of friction between the cylinder and the plane is equal to

k

. Find:

(a)

how long the cylinder will move with sliding;

(b)

the total work performed by the sliding friction force acting on the cylinder.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>DYNAMIC OF A SOLID BODY>Q 1.263.

A uniform ball of radius

r

, rolls, without slipping down, from the top of a sphere of radius

R

. Find the angular velocity of the ball at the moment it breaks off the sphere. The initial velocity of the ball is negligible.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>DYNAMIC OF A SOLID BODY>Q 1.264.

A uniform solid cylinder of radius $R = 15 cm$ , rolls over a horizontal plane, passing into an inclined plane, forming an angle $α = 30 °$ , with the horizontal. Find the maximum value of the velocity $v_{0}$ , which still permits the cylinder to roll onto the inclined plane section, without a jump. The sliding is assumed to be absent.

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$(Gravitational acceleration, g = 9.8 m s^{- 2})$

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>DYNAMIC OF A SOLID BODY>Q 1.265.

A small body $A$ is fixed to the inside of a thin rigid hoop, of radius $R$ and mass equal to that of the body $A$ . The hoop rolls, without slipping, over a horizontal plane. At the moment when the body $A$ gets into the lower position, the centre of the hoop moves with a velocity $v_{0}$ . At what values of $v_{0}$ , will the hoop move without bouncing?

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>DYNAMIC OF A SOLID BODY>Q 1.266.

Determine the kinetic energy of a tractor crawler belt of mass $m$ , if the tractor moves with a velocity $v$ .

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The arrangement shown in figure consists of two identical uniform solid cylinders, each of mass m, on which two light threads are wound symmetrically. Find the tension of each thread in the process of motion. The friction in the axle of the upper cylinder is assumed to be absent.

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

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The arrangement shown in figure consists of two identical uniform solid cylinders, each of mass $m$ , on which two light threads are wound symmetrically. Find the tension of each thread in the process of motion. The friction in the axle of the upper cylinder is assumed to be absent.