Engineering 100
Homework Problem #1
Below is a diagram of a beam supported on the left end by a pin connection, and toward the right side
by a simple fulcrum. A force F is applied to the far right end of the beam as shown. The uniform beam
has mass, m, length, l, and the distance from the left end to the fulcrum is d. Do the following:
1. Find the reaction forces at the pin and fulcrum in terms of F, m, l, d, and g, the
acceleration due to gravity.
2. Given m = 10 kg, l = 2m, d = 1.5m, and g = 9.81 m/s
2
, find F such that the reaction force
at the pin connection is zero.
To solve the problem, we need to analyze the forces acting on the beam and apply the principles of statics. Let's break down the problem into two parts:
1. Find the reaction forces at the pin and fulcrum in terms of F, m, l, d, and g:
Since the beam is in static equilibrium, the sum of forces and torques acting on it must be zero.
a) Reaction force at the pin connection:
- The weight of the beam (mg) acts downward at its center of mass, located at l/2 from the left end.
- The reaction force at the pin connection counters the vertical component of the weight.
Summing forces in the vertical direction:
R_pin - mg = 0 (Vertical equilibrium)
=> R_pin = mg
b) Reaction force at the fulcrum:
- The force F applied at the far right end of the beam acts downwards.
- The weight of the beam (mg) also contributes to the torque about the fulcrum.
Summing torques about the fulcrum:
- F * l - mg * (l - d) = 0 (Torque equilibrium)
=> F * l = mg * (l - d)
=> F = mg * (l - d) / l
2. Given m = 10 kg, l = 2m, d = 1.5m, and g = 9.81 m/s^2, find F such that the reaction force at the pin connection is zero:
To make the reaction force at the pin connection zero, R_pin = 0.
Using the equation from step 1a:
R_pin = mg = 0
=> mg = 0
=> 10 * 9.81 = 0
This equation is not possible. Therefore, there is no force F that can make the reaction force at the pin connection zero.
In conclusion, for the given values of m, l, d, and g, there is no force F that can make the reaction force at the pin connection zero.