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Chapter 5: Problem 56

At the end of a factory production line, boxes start from rest and slide down a \(30^{\circ}\) ramp 5.4 m long. If the slide can take no more than \(3.3 \mathrm{s}\), what's the maximum allowed frictional coefficient?

Short Answer

Expert verified
The maximum allowed frictional coefficient can be found by substituting the values given into the equation set up and solving for \( \mu \).

Step by step solution

01

Establish the kinematic equation

First, we establish the kinematic equation for the movement of the box down the ramp: \(d = v_i t + \frac{1}{2} a t^2\). Given that the box starts from rest, the initial velocity \(v_i\) is 0, and the distance \(d\) is 5.4 m. The time \(t\) is given as 3.3 s. We can solve for acceleration \(a\).
02

Solve for acceleration

Setting up the equation with the given values, we have: \(5.4 = 0 + \frac{1}{2}*a*(3.3^2)\). Solving for \(a\) gives us the acceleration in m/s².
03

Understand the forces on the box

The box moves under the effect of two forces - gravity and friction. The component of gravity acting along the ramp is \(g \sin(30^{\circ})\), and the frictional force equates to the frictional coefficient times the normal force, which is the component of gravity perpendicular to the ramp, or \(f = \mu \times (mg \cos(30^{\circ}))\). As the box doesn't accelerate upwards, the upward and downward forces must balance out (Newton's first law), thus the magnitude of the acceleration equals the frictional force divided by the box's mass.
04

Solve for the frictional coefficient

Setting \(a = g \sin(30^{\circ}) - \mu(g \cos(30^{\circ}))\), we can solve for \(\mu\). Given that we leave \(g\) in its standard form (9.8 m/s²), we can then find the maximum allowed frictional coefficient.

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Most popular questions from this chapter

Explain why a car with ABS brakes can have a shorter stopping distance.

You're in traffic court, arguing against a speeding citation. You entered a 210 -m-radius banked turn designed for \(80 \mathrm{km} / \mathrm{h},\) which was also the posted speed limit. The road was icy, yet you stayed in your lane, so you argue that you must have been going at the design speed. But police measurements show there was a frictional coefficient \(\mu=0.15\) between tires and road. Is it possible you were speeding, and if so by how much?

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