6.26 A 1 kg block situated on a rough incline is connected to a spring of spring constant 100 N m–1 as shown in Fig. 6.17. The block is released from rest with the spring in the unstretched position. The block moves 10 cm down the incline before coming to rest.
Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.
Net force acting on the block down on the incline = - F
= -
=mg ( )
At equilibrium,
Work done = Potential energy of the stretched string
mg ( ) = (1/2)kx2
1 x 10 x ( ) = (1/2) x 100 x 0.1
10 x (0.602 – = 0.5 x 100 x 0.1
<p><strong>6.26 </strong>Mass of the block = 1 kg</p><p>Spring constant = 100 N/m</p><p>Displacement of the block, x = 10 cm = 0.1 m</p><p>At equilibrium, normal reaction, R = <span title="Click to copy mathml"><math><mi>m</mi><mi>g</mi><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow></math></span></p><p>Frictional force, F = <span title="Click to copy mathml"><math><mi>? </mi><mi>R</mi></math></span> = <span title="Click to copy mathml"><math><mi>m</mi><mi>g</mi><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow></math></span></p><p>Net force acting on the block down on the incline = <span title="Click to copy mathml"><math><mi>m</mi><mi>g</mi><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow><mi></mi></math></span> - F</p><p>= <span title="Click to copy mathml"><math><mi>m</mi><mi>g</mi><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow></math></span> - <span title="Click to copy mathml"><math><mi>? </mi><mi>m</mi><mi>g</mi><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow></math></span></p><p>=mg ( <span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow><mo>-</mo><mi></mi><mi>? </mi><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow></math></span> )</p><p>At equilibrium, </p><p>Work done = Potential energy of the stretched string</p><p>mg ( <span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow><mo>-</mo><mi></mi><mi>? </mi><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow></math></span> ) = (1/2)kx2</p><p>1 x 10 x ( <span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow><mo>-</mo><mi></mi><mi>? </mi><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>? </mo><mrow><mn>37</mn><mo>°</mo></mrow></mrow></math></span> ) = (1/2) x 100 x 0.1</p><p>10 x (0.602 – <span title="Click to copy mathml"><math><mi></mi><mi>? </mi><mn>0.798</mn><mo>)</mo></math></span> = 0.5 x 100 x 0.1</p><p><span title="Click to copy mathml"><math><mi>? </mi><mo>=</mo><mn>0.128</mn></math></span></p>
This is a multiple choice answer as classified in NCERT Exemplar
(b) conserving energy between “O” ans ”A”
Ui + Ki = Uf + Kf
0+1/2mv2= mgh + 1/2mv’
(v’)2=v2-2gh = v’= ……….1
Let speed after emerging be v1 then
=1/2mv12=1/2[1/2mv’2]
1/2m(v1)2=1/4m(v’)2=1/4m[v2-2gh]
V1=………….2
From eqn 1 and 2
So v1 = v’/=v2(v’/2)
v1>v’/2
hence after emerging from the target velocity of the bullet is more than half of its earlier velocity v’
(d) as the velocity of the bullet changes to v’ which is less than v’ hence , path
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This is a multiple choice answer as classified in NCERT Exemplar
(b) conserving energy between “O” ans ”A”
Ui + Ki = Uf + Kf
0+1/2mv2= mgh + 1/2mv’
(v’)2=v2-2gh = v’= ……….1
Let speed after emerging be v1 then
=1/2mv12=1/2[1/2mv’2]
1/2m(v1)2=1/4m(v’)2=1/4m[v2-2gh]
V1=………….2
From eqn 1 and 2
So v1 = v’/=v2(v’/2)
v1>v’/2
hence after emerging from the target velocity of the bullet is more than half of its earlier velocity v’
(d) as the velocity of the bullet changes to v’ which is less than v’ hence , path, followed will change and the bullet reaches at point B instead of A
(f) as the bullet is passing through the target the loss in energy of the bullet is transferred to particles of the target . therefore their internal energy increases.
This is a multiple choice answer as classified in NCERT Exemplar
(b, d) When a man of mass m climbs up the staircases of height L, work done by the gravitational force on the man is mgl work done by internal muscular forces will be mgL as the change in kinetic is almost zero.
Hence total work done =-mgL + mgL=0
As the point of application of the contact forces does not move hence work done by reaction forces will be zero.
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