Skip to content
Exercises · 2.16

Q.Figure 2.13 gives the xx-tt plot of a particle executing one-dimensional simple harmonic motion. (You will learn about this motion in more detail in Chapter 13). Give the signs of position, velocity and acceleration variables of the particle at t=0.3t = 0.3 s, 1.21.2 s, −1.2-1.2 s.

Figure 2.13
Figure 2.13
Kerala DhseTextbookSubjective· 3mImportance★★★★★est
45% · 23/51 Questions
🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

The graph is x(t)=−Asin⁡(πt)x(t) = -A\sin(\pi t) with amplitude A>0A>0 and period 2 2\,s (it crosses zero at every integer tt and dips negative just after t=0t=0). Differentiating gives velocity and acceleration, and for SHM the acceleration always points opposite to the displacement. Evaluating the signs at the three instants gives (−,−,+)(-,-,+), (+,+,−)(+,+,-) and (−,+,+)(-,+,+).

Setting up the motion

The curve is zero at t=0t=0 and negative just after, so write

x(t)=−Asin⁡(ωt),ω=2πT=2π2 s=π rad s−1.x(t) = -A\sin(\omega t), \qquad \omega = \frac{2\pi}{T} = \frac{2\pi}{2\,\text{s}} = \pi\ \text{rad s}^{-1}.

Then

v(t)=dxdt=−Aπcos⁡(πt),a(t)=dvdt=Aπ2sin⁡(πt)=−ω2x(t).v(t) = \frac{dx}{dt} = -A\pi\cos(\pi t), \qquad a(t) = \frac{dv}{dt} = A\pi^2\sin(\pi t) = -\omega^2 x(t).

The last relation, a=−ω2xa = -\omega^2 x, is the defining property of SHM: acceleration is always opposite in sign to the position.

Evaluating the signs

At t=0.3 t = 0.3\,s (0.3π=54∘0.3\pi = 54^\circ):

  • x=−Asin⁡54∘=−0.81A<0x = -A\sin 54^\circ = -0.81A < 0 → negative
  • v=−Aπcos⁡54∘=−0.59Aπ<0v = -A\pi\cos 54^\circ = -0.59A\pi < 0 → negative
  • a=−ω2x>0a = -\omega^2 x > 0 → positive

At t=1.2 t = 1.2\,s (1.2π=216∘1.2\pi = 216^\circ):

  • x=−Asin⁡216∘=−A(−0.59)=+0.59A>0x = -A\sin 216^\circ = -A(-0.59) = +0.59A > 0 → positive
  • v=−Aπcos⁡216∘=−Aπ(−0.81)=+0.81Aπ>0v = -A\pi\cos 216^\circ = -A\pi(-0.81) = +0.81A\pi > 0 → positive
  • a=−ω2x<0a = -\omega^2 x < 0 → negative …
Start your 14-day free trial to unlock the full solution →

Unlock everything free for 14 days

  • Full step-by-step solutions
  • Concept-first explanations
  • Methods, shortcuts & mistakes
  • PYQ mapping + timed mock tests
Start 14-day free trial →See Plans & Pricing →

Full access for 14 days. No credit card required.

←Previous question2.15 — Suggest a suitable physical situation for each of the following graphs…→Next question2.17 — Figure 2.14 gives the x-t plot of a particle in one-dimensional motion…