Express your answer in terms of,,,,, and, the magnitude of the acceleration due to gravity



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Precarious Lunch

A uniform steel beam of length and mass is bolted to the side of a building. The beam is supported by a steel cable attached to the end of the beam at an angle , as shown. The wall exerts an unknown force, , on the beam. A workman of mass sits eating lunch a distance from the building.



  1. Find , the tension in the cable. Remember to account for all the forces in the problem.

    Express your answer in terms of , , , , , and , the magnitude of the acceleration due to gravity.

    =

    g*(m_1*L/2+m_2*d)/(L*sin(theta))










  2. Find , the x-component of the force exerted by the wall on the beam (), using the axis shown. Remember to pay attention to the direction that the wall exerts the force.

    Express your answer in terms of and other given quantities.

    =

    -T*cos(theta)










  3. Find , the y-component of force that the wall exerts on the beam (), using the axis shown. Remember to pay attention to the direction that the wall exerts the force.

    Express your answer in terms of , , , , and .

=

-sin(theta)*T+m_1*g+m_2*g










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Sliding Dresser


Sam is trying to move a dresser of mass and dimensions of length and height by pushing it with a horizontal force applied at a height above the floor. The coefficient of kinetic friction between the dresser and the floor is and is the magnitude of the acceleration due to gravity. The ground exerts upward normal forces of magnitudes and at the two ends of the dresser. Note that this problem is two dimensional.

  1. If the dresser is sliding with constant velocity, find , the magnitude of the force that Sam applies.

    Express the force in terms of , , and .

    =

    mu_k*m*g










  2. Find the magnitude of the normal force . Assume that the legs are separated by a distance , as shown in the figure.

    Express this normal force in terms of , , , , and .

    =

    m*g/2-mu_k*m*g*h/L










  3. Find the magnitude of the normal force . Assume that the legs are separated by a distance , as shown in the figure.

    Answer in terms of , , , , and .

    =

    m*g/2+mu_k*m*g*h/L










  4. Find , the maximum height at which Sam can push the dresser without causing it to topple over.

    Express your answer for the maximum height in terms and .

=

L/(2*mu_k)










A Person Standing on a Leaning Ladder


A uniform ladder with mass and length rests against a smooth wall. A do-it-yourself enthusiast of mass stands on the ladder a distance from the bottom (measured along the ladder). The ladder makes an angle with the ground. There is no friction between the wall and the ladder, but there is a frictional force of magnitude between the floor and the ladder. is the magnitude of the normal force exerted by the wall on the ladder, and is the magnitude of the normal force exerted by the ground on the ladder. Throughout the problem, consider counterclockwise torques to be positive. None of your answers should involve (i.e., simplify your trig functions).

  1. What is the minimum coeffecient of static friction required between the ladder and the ground so that the ladder does not slip?

    Express in terms of , , , , and .

    =

    (m_1*d+m_2*L/2)*cos(theta)/((m_1+m_2)*L*sin(theta))










  2. Suppose that the actual coefficent of friction is one and a half times as large as the value of . That is, . Under these circumstances, what is the magnitude of the force of friction that the floor applies to the ladder?

    Express your answer in terms of , , , , , and . Remember to pay attention to the relation of force and .

=

(m_1*d+m_2*L/2)*cos(theta)*g/(L*sin(theta))










Three-Legged Table


The top view of a table, with weight , is shown in the figure. The table has lost the leg at (, ), in the upper right corner of the diagram, and is in danger of tipping over. Company is about to arrive, so the host tries to stabilize the table by placing a heavy vase (represented by the green circle) of weight at (, ). Denote the magnitudes of the upward forces on the table due to the legs at (0, 0), (, 0), and (0, ) as , , and , respectively.

  1. Find , the magnitude of the upward force on the table due to the leg at (, 0).

    Express the force in terms of , , , , , and/or . Note that not all of these quantities may appear in the answer.

    =

    W_v*X/L_x + W_t/2










  2. Find , the magnitude of the upward force on the table due to the leg at (0, ).

    Express the force in terms of , , , , , and/or . Note that not all of these quantities may appear in the final answer.

    =

    W_v*Y/L_y+W_t/2










  3. Find , the magnitude of the upward force on the table due to the leg at (0, 0).

    Express the force in terms of , , , , , , , and/or . Note that not all terms may appear in the answer.

=

W_v*(1-X/L_x-Y/L_y)










While the host is greeting the guests, the cat (of weight ) gets on the table and walks until her position is .

  1. Find the maximum weight of the cat such that the table does not tip over and break the vase.
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