the bend moment M of a beam is given by dm/dx=-w(1-x) where w and l are constants.determine M in terms of x given that M=wl^2/2
plz show me working step
dm/dx = -w(l-x)
dm = -w(l-x) dx
m = w(l-x)^2/2 + C
you cannot just say that M = wl^2/2
It has to be for some value of x. I presume you meant to say that
M(0) = wl^2/2
in which case C=0
To determine M in terms of x, we have the differential equation:
dm/dx = -w(1-x)
Let's solve this differential equation step-by-step:
Step 1: Separate the variables
dm = -w(1-x) dx
Step 2: Integrate both sides with respect to their respective variables
∫ dm = ∫ -w(1-x) dx
Step 3: Integrate dm and simplify the right side
M = -w ∫ (1-x) dx
M = -w [x - x^2/2] + C
Step 4: Use the initial condition to solve for the constant C
Given M = wl^2/2, plug in the values and solve for C:
wl^2/2 = -w [l - (l^2/2)] + C
wl^2/2 = -wl + wl^2/2 + C
C = wl
Step 5: Substitute the value of C back into the equation for M
M = -w [x - x^2/2] + wl
Simplifying this expression, we get the final answer:
M = wx^2/2 - wx + wl
To find M in terms of x, we'll need to solve the differential equation dm/dx = -w(1-x). Let's go step by step to find the solution.
Step 1: Separate Variables
To start, we separate variables by moving all the terms containing x to one side and all the terms containing M to the other side of the equation:
dm = -w(1-x) dx
Step 2: Integrate both sides
Next, we integrate both sides of the equation with respect to their respective variables. Integrating dm gives us M and integrating -w(1-x) dx gives us a function in terms of x:
∫ dm = ∫ -w(1-x) dx
M = -w ∫ (1-x) dx
Step 3: Evaluate the integral
We can now integrate the function on the right side of the equation. Integrating (1-x) with respect to x results in:
M = -w [x - (1/2)x^2] + C
where C is the constant of integration.
Step 4: Solve for C (Using given condition)
To find the value of C, we'll use the given condition that M = wl^2/2. Plugging this into the equation, we get:
wl^2/2 = -w [l - (1/2)l^2] + C
Simplifying the equation:
wl^2/2 = -wl + (w/2)l^2 + C
Rearranging and combining like terms:
wl^2/2 + wl - (w/2)l^2 = C
Step 5: Final Expression
Finally, substituting the value of C back into the equation, we get the expression for M in terms of x:
M = -w [x - (1/2)x^2] + wl^2/2 + wl - (w/2)l^2
The bending moment M of the beam is given by dM/dx= -w(l-x) where w is a constant
Determine of M. Given M = ½ wt^2 when x=0