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Chapter 4: Problem 67

A particle of mass \(5 \mathrm{~kg}\) moving in the \(X-Y\) plane has its potential energy given by \(U=(-7 x+24 y)\) Joule. The particle is initially at origin and has velocity \(\vec{u}=(14.4 \hat{i}+4.2 \hat{j}) \mathrm{m} / \mathrm{s}\) (A) The particle has speed \(25 \mathrm{~m} / \mathrm{s}\) at \(t=4 \mathrm{~s}\). (B) The particle has an acceleration \(5 \mathrm{~m} / \mathrm{s}^{2}\). (C) The acceleration of particle is normal to its initial velocity. (D) None of these.

Short Answer

Expert verified
The correct answer is (D) None of these, since all of the statements A, B, and C are true.

Step by step solution

01

Find the force on the particle

The force on a particle can be found from the gradient of the potential energy function U(x, y). Using the given potential energy function, we can find the force components in the x and y directions: $$F_x = -\frac{dU}{dx}$$ $$F_y = -\frac{dU}{dy}$$ Computing the derivatives, we get: $$F_x = -(-7) = 7 $$ $$F_y = -(24) = -24 $$ Therefore, the force on the particle is given by: $$\vec{F} = 7\hat{i} - 24\hat{j} \mathrm{N}$$
02

Compute the acceleration

Now that we have the force on the particle, we can compute its acceleration using Newton's second law: $$\vec{a} = \frac{\vec{F}}{m}$$ Substituting the given mass and the computed force, we get: $$\vec{a} = \frac{7\hat{i} - 24\hat{j}}{5 \, \mathrm{kg}} = 1.4\hat{i} - 4.8\hat{j} \, \mathrm{m/s^2} $$ Since none of the given statements involve the position of the particle, we don't need to find its position as a function of time.
03

Check Statement A

To check if the particle has a speed of 25 m/s at t = 4 s, we first need to find the particle's velocity as a function of time. Since the acceleration is constant, we can use the following formula: $$\vec{v}(t) = \vec{u} + \vec{a}t$$ Substituting the given initial velocity, computed acceleration, and time t = 4 s, we get: $$\vec{v}(4) = (14.4\hat{i} + 4.2\hat{j}) + (1.4\hat{i} - 4.8\hat{j})4 = (20.0\hat{i} - 15.0\hat{j}) \, \mathrm{m/s}$$ Now we can find the speed of the particle at t = 4 s: $$|\vec{v}(4)| = \sqrt{(20.0)^2 + (-15.0)^2} = 25 \, \mathrm{m/s}$$ Statement A is correct.
04

Check Statement B

To check if the particle has an acceleration of 5 m/s虏, we compare it with the computed acceleration: Since the computed acceleration is: $$\vec{a} = 1.4\hat{i} - 4.8\hat{j} \, \mathrm{m/s^2} $$ And its magnitude is: $$|\vec{a}| = \sqrt{(1.4)^2 + (-4.8)^2} = 5 \, \mathrm{m/s^2}$$ Thus, statement B is also correct.
05

Check Statement C

To check if the acceleration is normal to the initial velocity, we compute the dot product of the initial velocity and acceleration vectors. If the dot product is zero, then the vectors are orthogonal: $$\vec{u} \cdot \vec{a} = (14.4\hat{i} + 4.2\hat{j}) \cdot (1.4\hat{i} - 4.8\hat{j}) = 14.4(1.4) + 4.2(-4.8) = 0$$ Since the dot product is zero, the acceleration is normal to the initial velocity. Statement C is correct. Given that all statements A, B, and C are correct, the correct answer is (D) None of these, since all of the statements are true.

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