Question

2. 2/10 points Previous Answers SerPSE10 5.AE.011. My Notes Experimental Determination of H's and The following is a simple method of measuring coefficients of friction. Suppose a block is placed on a rough surface inclined relative to the horizontal as shown in the figure. The incline angle is increased until the block starts to move. Show that you can obtain H by measuring the critical angle e, at which this slipping just occurs. The external forces exerted on a block lying on a rough incline are the gravitational force mg, the normal forcen and the force of friction fs. For convenience, the gravitational force is resolved into a component mg sin(0) along the incline and a component mg cos(9) perpendicular to the incline. mg cos/ coso SOLUTION Conceptualize Consider the figure and imagine that the block tends to slide down the incline due to the gravitational force. To simulate the situation, place a coin on a book's cover and tilt the book until the coin begins to slide. Notice how this example differs from the example "The Runaway Car." When there is no friction on an incline, angles less than 45° x of the incline will cause a stationary object to begin moving. When there is friction, however, there is no movement of the object for angles less than the critical angle. Categorize The block is subject to various forces. Because we are raising the plane to the angle at which the block is just ready to begin to move but is not moving, we categorize the block as a particle in equilibrium . Analyze the diagram in the figure shows the forces on the block: the gravitational force mg, the normal force n, and the force of static friction fs. We choose x to be parallel to the plane and y perpendicular to it. According to the figure, which component of the gravitational force is perpendicular to the incline? mg sin() mg cos(0) mg tan(0) From the particle in equilibrium model, apply F = 0 to the block in both the x- and y-directions. (Assume the +x-direction is down the plane and the +y-direction is perpendicularly up from the plane. For equation (1), use the following as necessary: g, m, 0, and fs, where the subscript is lowercase. For equation (2), use the following as necessary: g, m, n, and 6.) (1) Fx - mg sin(a) 0 (2) fyD- Substitute mg = las from Equation (2) into Equation (1): (3) fs = mg sin(O) = sin(6) = n tan(e) When the incline angle is increased until the block is on the verge of slipping, the force of static friction has reached its maximum value won. The angle 9 in this situation is the critical angle lowercase.) Msn = n tan() . Make these substitutions in Equation (3). (Use the following as necessary: n and where the subscript is We have shown, as requested, that the coefficient of static friction is related only to the critical angle. For example, if the block just slips at c = 20.0°, we find that's Finalize Once the block starts to move at 92 , it accelerates down the incline and the force of friction is f ? . If 9 is reduced to a value less than , however, it may be possible to find an angle 'such that the block moves down the incline with constant speed as a particle in equilibrium again (ex = 0). In this case, use Equations (1) and (2) with fs replaced by fk to find 4: 4k =tan(@') where ? EXERCISE

Answer 1

Zfx = mig sine-fx = 0 Ify = n- anglose = 0 ② fx = intane - Msn = ntan Be - us = tante