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Chapter 4: Problem 562

A particle is moving under the influence of a force given by \(\mathrm{F}=\mathrm{kx}\), where \(\mathrm{k}\) is a constant and \(\mathrm{x}\) is the distance moved. What energy (in joule) gained by the particle in moving from \(\mathrm{x}=1 \mathrm{~m}\) to \(\mathrm{x}=3 \mathrm{~m} ?\) (A) \(2 \mathrm{k}\) (B) \(3 \mathrm{k}\) (C) \(4 \mathrm{k}\) (D) \(9 \mathrm{k}\)

Short Answer

Expert verified
The energy gained by the particle in moving from \(x = 1m\) to \(x = 3m\) is (C) \(4k\).

Step by step solution

01

Recall the formula for work

The formula for work done by a force that varies with position is given by: \[W = \int_{x_1}^{x_2} F(x) dx\] In our case, \(F(x) = kx\), \(x_1 = 1m\) and \(x_2 = 3m\).
02

Calculate the work done by the force

Plug the values into the formula for work and calculate the integral: \[W = \int_{1}^{3} kx \,dx\] To integrate, use the power rule: \(\int x^n dx = \frac{1}{n+1} x^{n+1} + C\). In this case, \(n = 1\), so: \[W = k \left(\frac{1}{2} x^{2}\right)\Bigr|_{1}^{3}\]
03

Evaluate the integral

Now, evaluate the expression at the limits of integration: \[W = k\left[\frac{1}{2} (3^2) - \frac{1}{2} (1^2)\right] = k\left[\frac{9}{2} - \frac{1}{2} \right] = k\left[\frac{8}{2}\right] = 4k\] So, the work done by the force is \(4k\).
04

Calculate the energy gained by the particle

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy: \[W = \Delta KE\] In our case, we found that the work done, \(W = 4k\). Therefore, the energy gained by the particle is also equal to \(4k\). The correct answer is (C) \(4k\).

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Most popular questions from this chapter

If Wa, Wb, and Wc represent the work done in moving a particle from \(\mathrm{X}\) to \(\mathrm{Y}\) along three different path $\mathrm{a}, \mathrm{b}\(, and \)\mathrm{c}$ respectively (as shown) in the gravitational field of a point mass \(\mathrm{m}\), find the correct relation between \(\mathrm{Wa}, \mathrm{Wb}\) and \(\mathrm{Wc}\) (A) \(\mathrm{Wb}>\mathrm{Wa}>\mathrm{Wc}\) (B) \(\mathrm{Wa}<\mathrm{Wb}<\mathrm{Wc}\) (C) \(\mathrm{Wa}>\mathrm{Wb}>\mathrm{Wc}\) (D) \(\mathrm{Wa}=\mathrm{Wb}=\mathrm{Wc}\)

A metal ball of mass \(2 \mathrm{~kg}\) moving with a velocity of $36 \mathrm{~km} / \mathrm{h}$ has a head on collision with a stationary ball of mass \(3 \mathrm{~kg}\). If after the collision, the two balls move together, the loss in kinetic energy due to collision is (A) \(40 \mathrm{~J}\) (B) \(60 \mathrm{~J}\) (C) \(100 \mathrm{~J}\) (D) \(140 \mathrm{~J}\)

A spring is compressed by \(1 \mathrm{~cm}\) by a force of \(4 \mathrm{~N}\). Find the potential energy of the spring when it is compressed by \(10 \mathrm{~cm}\) (A) \(2 \mathrm{~J}\) (B) \(0.2 \mathrm{~J}\) (C) \(20 \mathrm{~J}\) (D) \(200 \mathrm{~J}\)

Assertion and Reason are given in following questions. Each question have four option. One of them is correct it. (1) If both assertion and reason and the reason is the correct explanation of the Assertion. (2) If both assertion and reason are true but reason is not the correct explanation of the assertion. (3) If the assertion is true but reason is false. (4) If the assertion and reason both are false. Assertion: Work done by centripetal force is zero. Reason: This is because centripetal force is always along the tangent. (A)1 (B) 2 (C) 3 (D) 4

A body is moved along a straight line by a machine delivering a constant power. The velocity gained by the body in time \(t\) is proportional to..... (A) \(t^{(3 / 4)}\) (B) \(t^{(3 / 2)}\) (C) \(t^{(1 / 4)}\) (D) \(t^{(1 / 2)}\)
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