Question

A single conservative force F(x) acts on a 1.0 kg
particle that moves along the x-axis. The potential
energy U(x) is given by
U (x) = 20 + (x - 2)2
Where x is in meters. At x = 5.0 m the particle
has a kinetic energy of 20 J.
The mechanical energy of the system is 7n. Find
n.​

Answer 1

Question :

A single conservative force F(x) acts on a 1.0 kg  particle that moves along the x-axis. The potential  energy U(x) is given by  U(x) = 20 + (x - 2)²  , where x is in meters. At x = 5.0 m the particle  has a kinetic energy of 20 J.  The mechanical energy of the system is 7n. Find  n.​

Solution :

With the action of single conservative force F(x) , a particle of mass 1 kg moves along the X-axis .

Potential energy , U(x) = 20 + (x - 2)²

where , x is in meters .

At , x = 5 m , the particle has a kinetic energy of 20 J .

At , x = 5 m , Potential energy is ,

➠ U(5) = 20 + (5 - 2)²

➠ U = 20 + (3)²

➠ U = 29 J

Given , Mechanical energy of the system is 7n J .

Mechanical energy is the addition of both potential and kinetic energies .

➠ 7n = 29 + 20

➠ 7n = 49

➠ n = 7 $\pink{\bigstar}$

Answer 2

$\Large{\orange{\underline{\textsf{\textbf{Solution\::}}}}}$

• On 1 kg particle a single conservative force F(x) acts and it moves along the x-axis.

➣ At x = 5 m, kinetic energy (K.E) of the particle is 20 J.

$\longmapsto\:\:\bf{Potential\:Energy\:\Big(U(x)\Big)\:=\:20\:+\:(x\:-\:2)^2\:}$

➣ At x = 5 m, potential energy is,

$\longmapsto\:\:\bf{U(5)\:=\:20\:+\:(5\:-\:2)^2\:}$

$\longmapsto\:\:\bf{U(5)\:=\:20\:+\:3^2\:}$

$\longmapsto\:\:\bf{U(5)\:=\:20\:+\:9\:}$

$\longmapsto\:\:\bf\blue{U(5)\:=\:29\:J}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ

$\red\bigstar\:\:{\underline{\green{\boxed{\bf{\purple{Mechanical\:Energy\:=\:U\:+\:K.E\:}}}}}}$

Aᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ǫᴜᴇsᴛɪᴏɴ,

☆ Mechanical energy = 7n

Tʜᴜs,

$:\implies\:\:\bf{7n\:=\:29\:+\:20\:}$

$:\implies\:\:\bf{7n\:=\:49\:}$

$:\implies\:\:\bf{n\:=\:\dfrac{49}{7}\:}$

$:\implies\:\:\bf\pink{n\:=\:7\:}$