Question

A 5kg block is sliding on a rough horizontal surface with speed 10m/s at the moment it compresses an uncompressed spring of spring constant 400 N/m .If the kinetic friction between the block and surface is 10N find how much has the spring compressed by:-

Answer 1

A block of mass 5kg is sliding on a rough horizontal surface with speed 10 m/s at the moment it compresses a spring of spring constant 400N/m.

kinetic friction between block and surface is, fr = 10N , then we have to find length of compression of spring.

from conservation of energy theorem ,

change in kinetic energy of block + workdone by frictional force = potential energy in spring.

or, 1/2mv² + fr × x cos180° = 1/2 kx²

[ frictional force acts just opposite of its motion so, angle between force and displacement is 180° ]

or, 1/2 mv² - fr × x = 1/2 kx²

or, 1/2 × 5 × (10)² - 10 × x = 1/2 × 400x²

or, 250 - 10x = 200x²

or, 20x² + x - 25 = 0

or, x = 1.093 m

hence, answer is 1.093 m

Answer 2

Answer:

1.09325 m

Step-by-step explanation:

Change in kinetic energy = Work done by all forces.

Change in Kinetic energy = Work done by the spring + Work done by the friction.

Given that the spring is compressed by x we have :

Work done by Spring = ½kx²

= 1/2 × 400 × x² = 200x²

Work done by Friction = F × x

= 10x

Change in kinetic energy = ½mv²

= ½ × 5 × 10² = 250

250 = 10x + 200x²

Divide through by 10:

25 = x + 20x²

We form a quadratic equation as follows :

20x² + x - 25 = 0

We solve using the quadratic formula :

{-1 +/- square root of [1² + 4 × 20 × 25]} / (20 × 2)

x = 43.73/40 or - 45.73/40

We take the positive value of x.

Therefore x = 43.73/40 = 1.09325 meters

The spring is compressed by 1.09325 m.