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Chapter 6: Problem 65

You do \(2.2 \mathrm{kJ}\) of work pushing a trunk at constant speed \(3.1 \mathrm{m}\) along a ramp inclined upward at \(22^{\circ} .\) What's the frictional coefficient between trunk and ramp?

Short Answer

Expert verified
The coefficient of friction (μ) between the trunk and the ramp is calculated from the given details using equations for work, equilibrium forces and friction. Please refer to the detailed calculation steps for the numerical value.

Step by step solution

01

Convert work from kJ to J

Given work (W) = 2.2 kJ, we need to convert it to joules (J).\nSo, \(W = 2.2 \times 1000 = 2200 J\)
02

Calculate the force applied

Since the trunk is moving at constant speed along the incline, this implies no acceleration. Thus, the force exerted is equal to the force due to friction. Next, we calculate these forces. The work done is equal to the frictional force times the displacement times the cosine of the angle between them. Given displacement \(s = 3.1 m\) and inclination \(θ = 22^\circ\) and using \(\cos(90^\circ) = 0\), we can write: \(W = Friction \times s\). Solving for frictional force, we get: \(Friction = W / s = 2200 / 3.1 \approx 709.68 N\)
03

Calculate normal force

Assuming the trunk's weight acts vertically downward, the component of the weight perpendicular to the ramp is the normal force. So, Normal Force = Weight x cos(θ). But here, the weight is not provided directly. However, since the trunk moves at a constant speed, the component of the weight parallel to the ramp equals the frictional force. This gives us: Weight x sin(θ) = Friction. Solving for weight, we have: Weight = Friction / sin(θ). Substituting the value of friction from Step 2 into this equation allows us to calculate the weight.
04

Calculate the coefficient of friction

The frictional force is also equal to the product of the coefficient of friction (μ) and the normal force. Using the equation Friction = μ x normal force and the fact that the normal force equals the component of the weight normal to the ramp from Step 3, we can now solve for μ. This gives us: μ = Friction / Normal Force. Substituting values calculated above gives us the coefficient of friction.

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