Question

a force of f = 85 n acts on the handle of the paper cutter at a as shown in (figure 1).
part a
if θ = 60°, determine the moment created by this force about the hinge at o.
express your answer to three significant figures and include the appropriate units. assume the positive direction is counterclockwise.
part b
at what angle θ should the force be applied so that the moment it creates about point o is a maximum (clockwise)?
express your answer in degrees to three significant figures.
part c

Explanation:

Step1: Recall moment formula

The moment of a force $M$ about a point is given by $M = rF\sin\alpha$, where $r$ is the distance from the point to the line - of - action of the force, $F$ is the magnitude of the force, and $\alpha$ is the angle between the position vector $\vec{r}$ and the force vector $\vec{F}$. Here, $r=(400 + 10)\text{mm}=410\text{mm}=0.41\text{m}$ and $F = 85\text{N}$.

Step2: Calculate moment for Part A

When $\theta = 60^{\circ}$, the angle between the position vector from $O$ to $A$ and the force vector is $\alpha=60^{\circ}+30^{\circ}=90^{\circ}$. Using the moment formula $M = rF\sin\alpha$, we substitute $r = 0.41\text{m}$, $F = 85\text{N}$, and $\sin\alpha=\sin90^{\circ}=1$. So $M=(0.41)\times(85)\times1 = 34.85\text{N}\cdot\text{m}\approx34.9\text{N}\cdot\text{m}$.

Step3: Analyze maximum moment condition for Part B

The moment $M = rF\sin\alpha$. Since $r$ and $F$ are constant, $M$ is maximum when $\sin\alpha$ is maximum. The maximum value of $\sin\alpha$ is 1, which occurs when $\alpha = 90^{\circ}$. Given the geometry of the problem, if the angle between the handle and the horizontal is $30^{\circ}$, then $\theta+30^{\circ}=90^{\circ}$, so $\theta = 60.0^{\circ}$.

Answer:

Part A: $34.9\text{N}\cdot\text{m}$
Part B: $60.0^{\circ}$