Concept explainers
A small 250-g collar C can slide on a semicircular rod which is made to rotate about the vertical AB at a constant rate of 7.5 rad/s. Knowing that the coefficients of friction are μs = 0.25 and μk = 0.20, indicate whether the collar will slide on the rod if it is released in the position corresponding to (a) θ = 75°, (b) θ = 40°. Also, determine the magnitude and direction of the friction force exerted on the collar immediately after release.
Fig. P12.64 and P12.65
(a)
Indicate whether the collar will slide on the rod if it is released in the position corresponding to angle
Find the magnitude and direction of the friction force exerted on the collar immediately after release.
Answer to Problem 12.65P
The collar does not slide on the rod if it is released in the position corresponding to angle
The magnitude and direction of the friction force exerted on the collar immediately after release is 0.611 N.
Explanation of Solution
Given information:
The mass of collar (m) is 250 g.
The speed of rotation of semi-circular rod
The radius (r) of semicircular rod is 500 mm.
The coefficient of static friction
The coefficient of kinetic friction
Calculation:
The collar will not slide. Therefore the collar moves at constant speed.
Find the equation of radius of circle for constant rotation.
Find the equation of speed of semicircular rod for constant rotation:
Find the equation of normal acceleration
Substitute
Substitute
Sketch the free body diagram and kinetic diagram of the collar C as shown in Figure (1).
Refer Figure (1):
Find the normal force (N) on the collar.
Apply Newton’s law of equation along x-axis.
Substitute
Substitute 250 g for m,
Find the frictional force (F) using Newton’s law of equation:
Apply Newton’s law of equation along y-axis.
Substitute
Substitute 250 g for m,
Find the frictional force using general equation:
Substitute 0.25 for
The frictional force
Thus, the magnitude and direction of the friction force exerted on the collar immediately after release is 0.611 N.
(b)
Indicate whether the collar will slide on the rod if it is released in the position corresponding to angle
Find the magnitude and direction of the friction force exerted on the collar immediately after release.
Answer to Problem 12.65P
The collar does not slide on the rod if it is released in the position corresponding to angle
The magnitude and direction of the friction force exerted on the collar immediately after release is 0.957 N.
Explanation of Solution
Calculation:
Find the normal force (N) on the collar using Equation (1):
Substitute 250 g for m,
Find the frictional force (F) using Equation (2):
Substitute 250 g for m,
Find the frictional force using general equation:
Substitute 0.25 for
The frictional force
Sketch the free body diagram and kinetic diagram of the collar C which is sliding as shown in Figure (2).
Refer Figure 2:
Find the normal force (N) on the collar.
Apply Newton’s law of equation along x-axis.
Substitute
Substitute 250 g for m,
Find the magnitude and direction of the friction force
Substitute 0.20 for
Thus, the magnitude and direction of the friction force exerted on the collar immediately after release is 0.957 N.
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