Lesson Using vectors to solve problems in Mechanics: Force

Using vectors to solve problems in Mechanics: Force

Problem 1

A 100-pound box is pulled at a constant speed on a horizontal surface by the force parallel to                             the floor  (Figure 1).  The friction coefficient is   = 0.2. Find the magnitude of the applied force. Solution At the given conditions,  the friction force has the magnitude F_frict = * Weight = 0.2*100 = 20 pounds. So,  the applied force should have the same magnitudes as the friction force,  i.e.  20 pounds providing the uniform move of the box. Answer.  The magnitude of the applied force is equal to  20  pounds.

Figure 1.  The body on the horizontal floor

Problem 2

A 100-pound box is pulled at a constant speed on a horizontal floor by the force of  20.7  pounds                       that makes an angle   = 30°  with the horizontal  (Figure 2).  Find the friction coefficient. Solution Decompose the given force  F  into the horizontal and vertical components. The horizontal component is equal to  |F|*cos(30°),  the vertical component is  |F|*sin(30°). The reaction force  N  acts from the floor vertically up to the box.  Together with the vertical component  Fy  it balances the weight:  |W| = Fy + |N|.  So,  |N| = |W| - Fy. The friction coefficient is the ratio of the shear force Fx to the reaction force  |N|: = = = = = 0.2.

Figure 2.  The body on the horizontal floor

Problem 3

The inclined plane makes an angle    with the horizontal as shown in  Figure 3.                                                       A  100-pound box stands at rest on the inclined plane kept by the friction force (force of static friction)  which is assumed large enough. Find the range of the angle    values where the force of static friction is over the shear force,  if the  static friction coefficient  is  . Solution The component of the weight normal to the inclined plane is   =  (normal force). The component of the weight parallel to the inclined plane is   =  (shear force). The force of static friction at these conditions is  .

Figure 3.  The body at rest on the inclined plane

Problem 4

A 100-pound box can move without friction on a  15°  ramp as shown in  Figure 5. You can imagine that the box is placed on a cart and the total weight of the box and the cart is  100  pounds. The cart is held on the ramp surface by a string,  which is tied to a peg on the top of the ramp,  so the cart and the box do not move.  The string is parallel to the ramp surface. Find the tension of the string. Solution This time the friction is absent,  according to the condition,  so we are not interested in calculating the normal force to the inclined surface.  All that we need to know is the                                          shear force only.

Figure 4.  The cart at rest on the inclined               plane fixed by a string

Problem 5

The inclined plane makes an angle    with the horizontal as shown in  Figure 5.                                                     A  100-pound box is pulled up on the inclined plane by the force  F  parallel to its surface. Find the magnitude of this force, if the  kinematic friction coefficient    is equal to 0.2. Solution The component of the weight normal to the inclined plane is   =  (normal force). The component of the weight parallel to the inclined plane is   =  (shear force). The force of kinematic friction at these conditions is  .

Figure 5.  The body pulled up on the inclined plane