Vectors and matrices — Question 16

QCAA 2022, Paper 2 · 6 marks

Q16 · 2022 · Technology-activeComplex familiar6 marks

QUESTION 16 (6 marks)

An object with a mass of 12 kg lies on a frictionless inclined plane. A rope is attached to the object at an angle of

25^\circ

above the plane, as shown. The force of the rope,

T

N, prevents the object from moving. When the rope is detached, the object moves down the plane with an acceleration of

5.6

m s

^{-2}

. Determine the magnitude of

T

WORKED SOLUTION

6 marks

ANSWER

T\approx74.15

Worked solution

With the rope attached, resolving parallel to the plane gives

|T|\cos25^\circ=12g\sin\theta.

When the rope is detached, the net force is

ma=12(5.6)=67.2

N and

12g\sin\theta=67.2,

\theta\approx34.85^\circ

. Substituting into the equilibrium equation gives

|T|\cos25^\circ=12g\sin34.85^\circ,

therefore

|T|\approx74.15

correctly determines an equation in terms of

|T|

(or

T

) and

\theta

when the rope is attached

[1 mark]

correctly calculates the magnitude of the net force acting when the rope is detached

[1 mark]

determines an equation in terms of

\theta

when the rope is detached

[1 mark]

determines a value representing

\theta

[1 mark]

determines an equation in terms of

|T|

(or

T

)

[1 mark]

determines the magnitude of

T

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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