SATHEE: Simple Harmonic Motion 4 Question 4

Simple Harmonic Motion 4 Question 4

4. A block of mass $m$ lying on a smooth horizontal surface is attached to a spring (of negligible mass) of spring constant $k$ . The other end of the spring is fixed as shown in the figure. The block is initially at rest in its equilibrium position. If now the block is pulled with a constant force $F$ , the maximum speed of the block is

(2019 Main, 09 Jan I)

(a) $\frac{π F}{\sqrt{m k}}$

(b) $\frac{F}{\sqrt{m k}}$

(d) $\frac{F}{π \sqrt{m k}}$

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Answer:

Correct Answer: 4. (b)

Solution:

In a spring-block system, when a block is pulled with a constant force $F$ , then its speed is maximum at the mean position. Also, it’s acceleration will be zero.

In that case, force on the system is given as,

$F = k x$

where, $x$ is the extension produced in the spring.

$or x = \frac{F}{k}$

Now we know that, for a system vibrating at its mean position, its maximum velocity is given as,

$v_{max} = A ω$

where, $A$ is the amplitude and $ω$ is the angular velocity.

Since, the block is at its mean position.

$So, \begin{aligned} A & = x = \frac{F}{k} \\ v_{max} & = \frac{F}{k} \sqrt{\frac{k}{m}} ∵ ω = \sqrt{\frac{k}{m}} = \frac{F}{\sqrt{k m}} \end{aligned}$

Alternate Method

According to the work-energy theorem, net work done $=$ change in the kinetic energy

Here, net work done $=$ work done due to external force $(W_{ext}) +$ work done due to the spring $(W_{spr})$ .

$\begin{aligned} As, W_{ext} = F \cdot x \\ and W_{s p r} = \frac{- 1}{2} k x^{2} \\ \Rightarrow Δ K E = F \cdot x + - \frac{1}{2} k x^{2} \\ (Δ K E)_{f} - (Δ K E)_{i} = F \cdot x - \frac{1}{2} k x^{2} \\ \Rightarrow \frac{1}{2} m v_{max}^{2} - \frac{1}{2} m (0)^{2} = F \cdot \frac{F}{k} - \frac{1}{2} k {\frac{F}{k}}^{2} \\ \Rightarrow \frac{1}{2} m v_{max}^{2} = \frac{F^{2}}{k} - \frac{F^{2}}{2 k} = \frac{F^{2}}{2 k} \\ or v_{max}^{2} = \frac{F^{2}}{k m} \\ \Rightarrow v_{max} = F / \sqrt{k m} \end{aligned}$

Simple Harmonic Motion 4 Question 4

Answer:

Alternate Method

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Simple Harmonic Motion 4 Question 4

Answer:

Alternate Method

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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