In Fig. 12-42, what magnitude of (constant) force applied horizontally at the axle of the wheel is necessary to raise the wheel over a step obstacle of height ? The wheel’s radius is , and its mass is .
The magnitude of (constant) force applied horizontally at the axle of the wheel is .
Obstacle of height
Massof the wheel
Wheel’s radius is ,
Here, at the moment when the wheel leaves the lower floor, the floor no longer exerts a force on it.As the wheel is raised over the obstacle, the only forces acting are the force Fapplied horizontally at the axle, the force of gravity mg acting vertically at the center of the wheel, and the force of the step corner, shown as the two components androle="math" localid="1661752228965" .We have to calculate horizontal and vertical component of the force
There are four forces are acting on the wheel. The applied force as given and the downward force that is due to its weight at the centre. Now if we consider a point at the surface of the wheel, there are two extra forces that are a horizontal to the left and a vertical force to the top.
As the wheel is continuously so the normal force no longer is valid. So we consider the case of net torque to be zero.
If the minimum force is applied the wheel does not accelerate, so both the total force and the total torque acting on it are zero.
We calculate the torque around the step corner as a pivot point in the second diagram (above right) and the third diagram indicates that the distance from the line of F to the corner is , where r is the radius of the wheel and h is the height of the step. The distance from the line of mg to the corner is
Now, applying the net torque as zero at the pivot point to meet our conditions of balancing the forces, we get that
The solution for F is
From equation (a), we can say that if the height of the pivot point is increased, then the force that must be applied also goes up. Below is the plot F/mg as a function of the ratio h/ r. The required force increases rapidly as h/ r —>1.
Hence, the magnitude of (constant) force applied horizontally at the axle of the wheel is .
A construction worker attempts to lift a uniform beam off the floor and raise it to a vertical position. The beam is long and weighs . At a certain instant the worker holds the beam momentarily at rest with one end at distance above the floor, as shown in Fig. 12-75, by exerting a force on the beam, perpendicular to the beam. (a) What is the magnitude P? (b) What is the magnitude of the (net) force of the floor on the beam? (c) What is the minimum value the coefficient of static friction between beam and floor can have in order for the beam not to slip at this instant?
In Fig. 12-68, an 817 kg construction bucket is suspended by a cable A that is attached at O to two other cables B and C, making angles and with the horizontal. Find the tensions in (a) cable A, (b) cable B, and (c) cable C. (Hint: To avoid solving two equations in two unknowns, position the axes as shown in the figure.)
Figure 12-84 shows a stationary arrangement of two crayon boxes and three cords. Box A has a mass of and is on a ramp at angle box B has a mass of and hangs on a cord. The cord connected to box A is parallel to the ramp, which is frictionless. (a) What is the tension in the upper cord, and (b) what angle does that cord make with the horizontal?
In Fig12-46, a uniform square sign, of edge length , is hung from a horizontal rod of length and negligible mass. A cable is attached to the end of the rod and to a point on the wall at distance above the point where the rod is hinged to the wall.(a) What is the tension in the cable? What are the (b) magnitude and) of the horizontal component of the force on the rod from the wall, and the (c) direction (left or right) of the horizontal component of the force on the rod from the wall, and the (d) magnitude of the vertical component of this force? And (e) direction (up or down) of the vertical component of this force?
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