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Chapter 5: Problem 18

A student wants to determine the coefficients of static friction and kinetic friction between a box and a plank. She places the box on the plank and gradually raises one end of the plank. When the angle of inclination with the horizontal reaches \(28.0^{\circ}\), the box starts to slip and slides \(2.53 \mathrm{~m}\) down the plank in \(3.92 \mathrm{~s}\). Find the coefficients of friction.

Short Answer

Expert verified
The coefficients of static and kinetic friction are approximately 0.53 and 0.49 respectively.

Step by step solution

01

Determination of static friction

First, find the coefficient of static friction. This can be determined by equating the gravitational force component acting parallel to the incline to the force of static friction. The force due to gravity acting down the incline is \( mg\sin{\theta} \) and the maximum static friction force is \( \mu_{s} mg\cos{\theta} \). At the point of slip, these two forces are equal. Solving \( \mu_{s} = \tan{\theta} \) for \(\mu_{s}\) and substituting \(\theta = 28.0^{\circ}\), we get \(\mu_{s} \approx 0.53\).
02

Determination of kinetic friction

Next, find the coefficient of kinetic friction. When the box is sliding, the force of kinetic friction balances the component of gravitational force along the incline. These must be equal for constant velocity. We can use the equation of motion that relates distance, time, initial velocity and acceleration to find the acceleration: \( x = ut + \frac{1}{2} a t^2 \), where \( u \) is initial velocity (zero in this case), \( x \) is distance slid (2.53m), and \( t \) is time (3.92s). Solving for acceleration, we get \( a \approx 0.327 m/s^2 \). Equating the net force \( mg\sin{\theta} - \mu_{k}mg\cos{\theta} \) to \( ma \), and solving for \( \mu_{k} \), we find that \( \mu_{k} \approx 0.49 \).
03

Solution

The coefficients of static and kinetic friction between the box and the plank are approximately \( \mu_{s} = 0.53 \) and \( \mu_{k} = 0.49 \), respectively.

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Most popular questions from this chapter

A circular curve of highway is designed for traffic moving at \(60 \mathrm{~km} / \mathrm{h}(=37 \mathrm{mi} / \mathrm{h}) .(a)\) If the radius of the curve is \(150 \mathrm{~m}\) \((=490 \mathrm{ft})\), what is the correct angle of banking of the road? ( \(b\) ) If the curve were not banked, what would be the minimum coefficient of friction between tires and road that would keep traffic from skidding at this speed?

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A \(136-\mathrm{kg}\) crate is at rest on the floor. A worker attempts to push it across the floor by applying a 412-N force horizontally. (a) Take the coefficient of static friction between the crate and floor to be \(0.37\) and show that the crate does not move. (b) A second worker helps by pulling up on the crate. What minimum vertical force must this worker apply so that the crate starts to move across the floor? \((c)\) If the second worker applies a horizontal rather than a vertical force, what minimum force, in addition to the original 412-N force, must be exerted to get the crate started?

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