Question 9
Three forces, $(2\hat{\boldsymbol{\imath}}+3\hat{\boldsymbol{\jmath}})$ N, $(5\hat{\boldsymbol{\imath}}-\hat{\boldsymbol{\jmath}})$ N and $(-\hat{\boldsymbol{\imath}}+6\hat{\boldsymbol{\jmath}})$ N, act concurrently on a particle of mass 2 kg. No other forces act on the particle.
The acceleration (m s$^{-2}$) of the particle is
That’s it. Step through the hints below to see why.
Not quite. $6\hat{\boldsymbol{\imath}}+8\hat{\boldsymbol{\jmath}}$ is the resultant force from hint 1. You stopped before dividing by the mass (2 kg).
Not quite. You multiplied the resultant by the mass instead of dividing, $2(6\hat{\boldsymbol{\imath}}+8\hat{\boldsymbol{\jmath}})=12\hat{\boldsymbol{\imath}}+16\hat{\boldsymbol{\jmath}}$.
Not quite. You divided the resultant by 3 (the number of forces, as if averaging them) instead of by the mass, $\frac{6\hat{\boldsymbol{\imath}}+8\hat{\boldsymbol{\jmath}}}{3}=2\hat{\boldsymbol{\imath}}+\frac{8}{3}\hat{\boldsymbol{\jmath}}$.
Blue: the three forces, tip to tail. Black: the resultant force $\boldsymbol{F}$. Dashed orange: the acceleration $\boldsymbol{a}$ from the option you are testing.
Find the resultant force by adding the three forces.
$$\begin{aligned}\hat{\boldsymbol{\imath}}\text{ components: }&\ 2+5+(-1)=6\\\hat{\boldsymbol{\jmath}}\text{ components: }&\ 3+(-1)+6=8\end{aligned}$$
$$\boldsymbol{F}=(6\hat{\boldsymbol{\imath}}+8\hat{\boldsymbol{\jmath}})\text{ N}$$
Newton’s second law uses the net (resultant) force. Concurrent forces act at the same point, so their vector sum is the net force. Add the $\hat{\boldsymbol{\imath}}$ components and the $\hat{\boldsymbol{\jmath}}$ components separately.
Apply Newton’s second law.
$$\boldsymbol{a}=\frac{\boldsymbol{F}}{m}$$
$\boldsymbol{F}=m\boldsymbol{a}$ links the net force to the acceleration. Rearrange for $\boldsymbol{a}$.
Substitute and divide each component by the mass.
$$\boldsymbol{a}=\frac{6\hat{\boldsymbol{\imath}}+8\hat{\boldsymbol{\jmath}}}{2}=3\hat{\boldsymbol{\imath}}+4\hat{\boldsymbol{\jmath}}$$
Dividing a vector by a scalar divides each component.
The resultant force is $(6\hat{\boldsymbol{\imath}}+8\hat{\boldsymbol{\jmath}})$ N. Dividing by the 2 kg mass gives the acceleration $(3\hat{\boldsymbol{\imath}}+4\hat{\boldsymbol{\jmath}})$ m s$^{-2}$.
$$\boldsymbol{a}=\frac{\boldsymbol{F}}{m}=\frac{6\hat{\boldsymbol{\imath}}+8\hat{\boldsymbol{\jmath}}}{2}=3\hat{\boldsymbol{\imath}}+4\hat{\boldsymbol{\jmath}}$$
Only mark this done when you could do it without help. Reading the solution does not count.
A Tangent Tuition mock exam question. It is not a past QCAA question, and Tangent Tuition is not affiliated with the QCAA.
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