A particle is moving in simple harmonic motion about the origin, with displacement $x$ metres. The displacement is given by $x = 2 ext{ sin } 3t$, where $t$ is time in seconds. The motion starts when $t = 0$. (i) What is the total distance travelled by the particle when it first returns to the origin? (ii) What is the acceleration of the particle when it is first at rest? The region bounded by $y = ext{cos } 4x$ and the $x$-axis, between $x = 0$ and $x = rac{ ext{π}}{2}$, is rotated about the $x$-axis to form a solid. Find the volume of the solid. A particle moves along a straight line with displacement $x$ m and velocity $v$ m s$^{-1}$. The acceleration of the particle is given by $rac{d^{2}x}{dt^{2}} = 2 - rac{x}{2}$. Given that $v = 4$ when $x = 0$, express $v^{2}$ in terms of $x$.

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A particle is moving in simple harmonic motion about the origin, with displacement $x$ metres - HSC - SSCE Mathematics Extension 1 - Question 12 - 2014 - Paper 1

Question 12

A particle is moving in simple harmonic motion about the origin, with displacement $x$ metres. The displacement is given by $x = 2 ext{ sin } 3t$, where $t$ is time... show full transcript

Worked Solution & Example Answer:A particle is moving in simple harmonic motion about the origin, with displacement $x$ metres - HSC - SSCE Mathematics Extension 1 - Question 12 - 2014 - Paper 1

Step 1

What is the total distance travelled by the particle when it first returns to the origin?

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Answer

To find the distance travelled by the particle when it first returns to the origin, we need to determine the period of the motion. The general formula for simple harmonic motion displacement is given by:

$x(t) = A ext{ sin }( heta)$

This implies that when the particle returns to the origin, the angle $heta$ must equal $0$ or $ext{π}$ , for the first complete return.

Given that $x = 2 ext{ sin }(3t)$ , we set:

$2 ext{ sin }(3t) = 0$

The first return to the origin occurs at:

$3t = 0 ext{ or } ext{π}$

Thus, the first return occurs at:

At this time, we calculate the total distance travelled. Within half a period:

From $t = 0$ to $t = rac{ ext{π}}{3}$ , the displacement goes from $0$ to $2$ m and back to $0$ m.
Therefore, the distance is: $|2| + |0| = 2 + 2 = 4 ext{ m}$ .

Step 2

What is the acceleration of the particle when it is first at rest?

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Answer

The particle is at rest when its velocity is zero. The velocity of the particle can be found by differentiating the displacement function:

$v(t) = rac{dx}{dt} = 6 ext{ cos }(3t).$

Setting this equal to zero to find when the particle is at rest:

$6 ext{ cos }(3t) = 0 \ \ ext{ implies } 3t = rac{ ext{π}}{2} \ \ ext{ or } 3t = rac{3 ext{π}}{2}$

This gives:

$t = rac{ ext{π}}{6} \ \ ext{ or } t = rac{ ext{π}}{2}$

The first instance occurs at $t = rac{ ext{π}}{6}$ . Now, substitute back to find the acceleration:

$a(t) = rac{d^{2}x}{dt^{2}} = -18 ext{ sin }(3t).$

Thus, when $t = rac{ ext{π}}{6}$ :

$$aigg(rac{ ext{π}}{6}igg) = -18 ext{ sin }igg(3 imes rac{ ext{π}}{6}igg) = -18 ext{ sin}igg(rac{ ext{π}}{2}igg) = -18 ext{ (1)} = -18 ext{ m/s}^{2}.$

Step 3

Find the volume of the solid.

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Answer

To find the volume of the solid formed by rotating the region bounded by $y = ext{cos} (4x)$ , the $x$ -axis between $x = 0$ and $x = rac{ ext{π}}{2}$ , we use the disk method:

$$V = ext{π} ext{ } igg[ ext{ Area integral}igg] = ext{π} ext{ } igg( ext{Area under } y = ext{cos}(4x)igg) = ext{π} ext{ }igg( igg[ rac{1}{2} y^2 igg]_{0}^{rac{ ext{π}}{2}} \ = ext{π} imes igg[ rac{1}{2}( ext{cos}(4 imes rac{ ext{π}}{2}))^2 - rac{1}{2}( ext{cos}(4 imes 0))^2\ = ext{π} imes igg[ rac{1}{2}(0^2) - rac{1}{2}(1^2) \ = -rac{ ext{π}}{2} \ \ V = ext{π} igg[ rac{1}{2}(1 - 0) = rac{ ext{π}}{2} ext{ } ( ext{Final positive volume})igg] = rac{ ext{π}}{2} ext{ m}^{3}.$

Step 4

Given that $v = 4$ when $x = 0$, express $v^{2}$ in terms of $x$.

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Answer

The acceleration is given by:

$rac{d^{2}x}{dt^{2}} = 2 - rac{x}{2}$

To find $v^2$ in terms of $x$ , we apply the chain rule:

$rac{dv}{dt} = rac{dv}{dx} imes rac{dx}{dt}$

This means:

$v rac{dv}{dx} = 2 - rac{x}{2}$

Integrating:

$rac{1}{2} v^2 = 2x - rac{x^2}{4} + C$

Given that $v = 4$ when $x = 0$ :

We can find $C$ . Therefore, $C = 8$ , giving:

$v^2 = 8 + 8 = 16$

Thus, the complete expression for $v^2$ is:

$$v^2 = 2x - rac{x^2}{4} + 16.$

A particle is moving in simple harmonic motion about the origin, with displacement $x$ metres - HSC - SSCE Mathematics Extension 1 - Question 12 - 2014 - Paper 1

Worked Solution & Example Answer:A particle is moving in simple harmonic motion about the origin, with displacement $x$ metres - HSC - SSCE Mathematics Extension 1 - Question 12 - 2014 - Paper 1

What is the total distance travelled by the particle when it first returns to the origin?

What is the acceleration of the particle when it is first at rest?

Find the volume of the solid.

Given that $v = 4$ when $x = 0$, express $v^{2}$ in terms of $x$.

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