The kinetic energy of a particle moving along a circle of radius R depends on the distance covered s as T = a s 2 , where a is a constant. Find the force acting on the particle as a function of s .

From the question, we have, The kinetic energy of the particle moving along a circle of radius R depends on the distance covered s as T = a s 2 , where a is a constant. Hence, T = 1 2 m v 2 = a s 2 . ⇒ v 2 = 2 a s 2 m . Differentiating the above equation both sides with respect to time we will get, 2 v d v d t = 4 a s m × d s d t ⇒ 2 v w t = 4 a s m v ⇒ w t = 2 a s m , where w t is tangential acceleration. On a circular path radial or centripetal acceleration = w n = v 2 R . Hence, the net acceleration w of the particle, w = w t 2 + w n 2 = 2 a s m 2 + 2 a s 2 m R 2 = 2 a s m 1 + s R 2 Then total force = F = m w = 2 a s 1 + s R 2 .

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A locomotive of mass $m$ starts moving so that its velocity varies according to the law $v = a \sqrt{s}$ , where $a$ is a constant, and $s$ is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first $t$ seconds after the beginning of motion.

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.120.

The kinetic energy of a particle moving along a circle of radius

R

depends on the distance covered

s

T = a s^{2}

, where

a

is a constant. Find the force acting on the particle as a function of

s

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.121.

A body of mass $m$ was slowly hauled up the hill by a force $\vec{F}$ which at each point was directed along a tangent to the trajectory. Find the work performed by this force, if the height of the hill is $h$ , the length of its base $W_{net} = ∆ K \Rightarrow W_{net} = \frac{1}{2} m v^{2} - 0$ , and the coefficient of friction is $k$ .

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.122.

A disc of mass

m = 50 g

slides with the zero initial velocity down an inclined plane set at an angle

α = 30 °

to the horizontal; having traversed the distance

l = 50 cm

along the horizontal plane, the disc stops. Find the work performed by the friction forces over the whole distance, assuming the friction coefficient

k = 0.15

for both inclined and horizontal planes.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.123.

Two bars of masses

m_{1}

and

m_{2}

connected by a non-deformed light spring rest on a horizontal plane. The coefficient of friction between the bars and the surface is equal to

k

. What minimum constant force has to be applied in the horizontal direction to the bar of mass

m_{1}

in order to shift the other bar?

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IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.124.

A chain of mass

m = 0.80 kg

, and length

l = 1.5 m

rests on a rough-surfaced table so that one of its ends hangs over the edge. The chain starts sliding off the table all by itself provided the overhanging part equals

η = \frac{1}{3}

of the chain length. What will be the total work performed by the friction forces acting on the chain by the moment it slides completely off the table?

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IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.125.

A body of mass

m

is thrown at an angle

α

to the horizontal with the initial velocity

v_{0}

. Find the mean power developed by gravity over the whole time of motion of the body, and the instantaneous power of gravity as a function of time.

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Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.126.

A particle of mass

m

moves along a circle of radius

R

with a normal acceleration varying with time as

w_{n} = a t^{2}

, where

a

is a constant. Find the time dependence of the power developed by all the forces acting on the particle, and the mean value of this power averaged over the first

t

seconds after the beginning of motion.

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IMPORTANT

Problems in General Physics>Chapter 1 - PHYSICAL FUNDAMENTALS OF MECHANICS>LAWS OF CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM>Q 1.127.

A small body of mass $m$ is located on a horizontal plane at the point $O$ . The body acquires a horizontal velocity $v_{0}$ . Find,

$(a)$ The mean power developed by the friction force during the whole time of motion, if the friction coefficient $k = 0.27, m = 1.0 kg$ , and $v_{0} = 1.5 m s^{- 1}$ ,

$(b)$ The maximum instantaneous power developed by the friction force, if the friction coefficient varies as $k = α x$ , where $α$ is a constant, and $x$ is the distance from the point $O$ .

A locomotive of mass m starts moving so that its velocity varies according to the law v=as, where a is a constant, and s is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first t seconds after the beginning of motion.

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

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