Newtonian Mechanics — Question 10

Question 10

A constraint equation must hold throughout the motion

Two points move on one line joined by an ideal rigid link of length 2 m, with B always to the right of A. A proposed history is $x_A=t^2$ m and $x_B=2+3t$ m, with numerical t in seconds.

1. Write the geometric constraint and its velocity consequence.

2. Test the proposed histories at 0 and 1 s.

3. Give a history for B compatible with A for all times, and find its acceleration.

Question 10 - Solution

A constraint equation must hold throughout the motion

(a) Differentiate a fixed separation.

$$x_B-x_A=L\ \Rightarrow\ v_B-v_A=0\ \Rightarrow\ a_B-a_A=0$$ The constraint is $x_B-x_A=2$ m, so $v_B-v_A=0$. Differentiating once more gives equal accelerations along this fixed line.

(b) Check more than initial geometry.

$$d(t)=2+3t-t^2:\quad d(0)=2,\ d(1)=4\,\mathrm m,\quad\dot d(0)=3\,\mathrm{m/s}$$ The initial separation is 2 m, but at 1 s it is $5-1=4$ m. Even at t=0 the velocities, 3 and 0 m/s, violate the derivative constraint. The proposed pair is not a rigid-link history.

(c) Enforce the boundary condition identically.

$$\boxed{x_B=2+t^2,\quad v_B=2t,\quad a_B=2\,\mathrm{m/s^2}}$$ $x_B=2+t^2$ m is compatible, giving $v_B=2t$ m/s and $a_B=2$ m/s$^2$. Geometric constraints reduce the independent variables before the dynamical equations are solved.