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 kg 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/s when it reaches the bottom of the ramp. You are given these coefficients of sliding (kinetic) friction for different pairs of materials.

Material 1 Material 2 Coefficient of Sliding Friction
Cast iron Cast iron 0.15
Cast iron Copper 0.29
Cast iron Lead 0.43
Cast iron Zinc 0.85

Answer 1

For cast iron we have

`h = 0.92 m`

For copper

`h = 1.05 m`

For Lead

`h = 1.23 m`

For Zinc

`h = 2.43 m`

Step-by-step explanation:

As we know that final speed of the block is calculated by work energy theorem

`W_f + W_g = (1)/(2)mv^2`

now we have

`-\mu_k mg cos\theta((h)/(sin\theta)) + mgh = (1)/(2)mv^2`

now we have

`v^2 = 2gh - 2\mu_k g h cot\theta`

`v = √(2gh(1 - \mu_k cot\theta))`

For cast iron we have

`4 = √(2(9.81)(h)(1 - 0.15cot52))`

`h = 0.92 m`

For copper

`4 = √(2(9.81)(h)(1 - 0.29cot52))`

`h = 1.05 m`

For Lead

`4 = √(2(9.81)(h)(1 - 0.43cot52))`

`h = 1.23 m`

For Zinc

`4 = √(2(9.81)(h)(1 - 0.85cot52))`

`h = 2.43 m`