Experimental Determination of Hs 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 0, 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 force n and the force of friction f.. For convenience, the gravitational force is resolved into a component mg sin(0) along the incline and a component mg cos(0) perpendicular to the incline. mg sin 0 mg cos 0 mg 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, ---Select--- v 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 ---Select--- 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 f.. 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? O mg tan(0) O mg sin(0) O mg cos(0)
Experimental Determination of Hs 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 0, 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 force n and the force of friction f.. For convenience, the gravitational force is resolved into a component mg sin(0) along the incline and a component mg cos(0) perpendicular to the incline. mg sin 0 mg cos 0 mg 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, ---Select--- v 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 ---Select--- 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 f.. 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? O mg tan(0) O mg sin(0) O mg cos(0)
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Question
Transcribed Image Text:Experimental Determination of
Hs
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 u. by measuring the critical angle 0, 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 force n and the force of friction f. For
convenience, the gravitational force is resolved
into a component mg sin(0) along the incline and
a component mg cos(0) perpendicular to the
incline.
mg sin e
mg cos 0
mg
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, ---Select---
v 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
---Select---
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 f.. 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?
O mg tan(0)
O mg sin(0)
O mg cos(0)
Transcribed Image Text: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 0.)
(1) Σ-
(2) Σ
= 0
Substitute mg =
from Equation (2) into Equation (1):
cos(0)
(3) f = mg sin(0) =
cos(e) sin(0) = 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 µ̟n. The angle 0 in this situation is the critical angle 0. Make these substitutions in Equation (3).
(Use the following as necessary: n and 0, where the subscript is lowercase.)
Hn = n tan(0)
Hs =
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 0, = 20.0°, we find that H, =
Finalize Once the block starts to move at 0 2 0, it accelerates down the incline and the force of friction is
f. ? v Hn. If 0 is reduced to a value less than 0, however, it may be possible to find an angle 0', such that the
block moves down the incline with constant speed as a particle in equilibrium again (a, = 0). In this case, use
Equations (1) and (2) with f, replaced by f, to find Hy: Hy = tan(8') where 0'. ? v 0.
EXERCISE
An untethered block sits on a flatbed truck as it accelerates up an incline that makes an angle of 15° with respect to
the horizontal. If the truck speeds up at a rate less than 3.90 m/s?, the block remains on the truck. If the truck
speeds up at a rate equal to or greater than this value, however, the block slides off the truck. What is the coefficient
of static friction between the truck and the block?
Hint
Hs =
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