Question

A long ramp made of cast iron is sloped at a constant angle θ = 52.0∘∘ above the horizontal. Small blocks, each with mass 0.42 kgkg but made of different materials, are released from rest at a vertical height h above the bottom of the ramp. In each case the coefficient of static friction is small enough that the blocks start to slide down the ramp as soon as they are released. You are asked to find h so that each block will have a speed of 4.00 m/sm/s when it reaches the bottom of the ramp. You are given these coefficients of sliding (kinetic) friction for different pairs of materials.(a) Use work and energy considerations to find the required value of h if the block is made from (i) cast iron; (ii) copper; (iii) zinc. (b) What is the required value of h for the copper block if its mass is doubled to 0.84 kg? (c) For a given block, if θ is increased while h is kept the same, does the speed v of the block at the bottom of the ramp increase, decrease, or stay the same?

Answer 1

Step-by-step explanation:

mass of block m = .42 kg .

coefficient of friction ( iron - iron) μ₁ = .15

coefficient of friction ( copper - iron) μ₂ = .29

coefficient of friction ( zinc - iron) μ₃ = .21

applying work energy principle

mgh - mμ g cosθ d = 1/2 mv² , d is length of ramp ,

gh - μ g cosθ d = 1/2 v²

gh - μ g cosθ h / sinθ = 1/2 v²

gh - μ g hcotθ = 1/2 v²

h = .5 v² / ( g - μ g cotθ)

1 ) For cast iron , for iron μ = .15

h = .5 v² / ( g - μ g cotθ)

= .5 x 4² / (g - .15 x 9.8 cot52)

= 8 / (9.8 - 1.148 )

= .9246 m

2 ) For copper , μ = .29

h = .5 v² / ( g - μ g cotθ)

= .5 x 4² / (g - .29 x 9.8 cot52)

= 8 / (9.8 - 2.22 )

= 1.055 m

3 ) For zinc , μ = .21

h = .5 v² / ( g - μ g cotθ)

= .5 x 4² / (g - .21 x 9.8 cot52)

= 8 / (9.8 - 1.607 )

= .9 758m

b )

Since height does not depend on mass , so height will remain same.

c )

h = .5 v² / ( g - μ g cotθ)

.5 v² = h x ( g - μ g cotθ)

if θ is increased , cotθ will decrease , ( g - μ g cotθ) will increase so v will increase.