A particle is constrained to move at a constant speed on an inclined plane (ABCD) along the curved path shown in the figure. Edges AD and BC are parallel to the y axis. The inclined plane makes an angle $\theta$ with the xy-plane. The velocity vector of the particle makes an angle $\phi$ with the dotted line which is parallel to edge AB. If the speed of the particle is $2$ m/s, $\phi = 30^\circ$, and $\theta = 40^\circ$, then the z-component of the velocity of the particle in m/s is _____________.
The particle moves at a constant speed on the inclined plane. We are required to find the z-component of the velocity vector. Given:
- Speed of the particle, \(v = 2 \, \text{m/s}\)
- Angle \(\phi = 30^\circ\)
- Inclined plane angle \(\theta = 40^\circ\)
The velocity vector makes an angle \(\phi\) with the dashed line parallel to edge AB.
The z-component of the velocity vector can be derived using the inclination angle \(\theta\). The relation for the z-component of velocity \(v_z\) is:
\({v_z} = v \cdot \sin\theta \cdot \cos\phi\)
Substituting the given values:
\({v_z} = 2 \cdot \sin(40^\circ) \cdot \cos(30^\circ)\)
Calculating each term:
- \(\sin(40^\circ) \approx 0.6428\)
- \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660\)
Thus,
\({v_z} = 2 \cdot 0.6428 \cdot 0.8660 = 1.114\)
The calculated value is approximately: \({v_z} \approx 1.11 \, \text{m/s}\) in the negative z-direction, as inferred from the diagram.
Thus, the z-component of the velocity of the particle is -1.11 m/s.
Important Questions from Vector Algebra
What is the length of projection of the vector \(\rm \hat{i}+2 \hat{j}+3 \hat{k}\) on the vector \(\rm2 \hat{i}+3 \hat{j}-2 \hat{k}\) ?
Consider the following in respect of the vectors \(\rm \vec{a}=(0,1,1)\) and \(\rm \vec{b}=(1,0,1) \) :
1. The number of unit vectors perpendicular to both \(\rm \vec{a}\) and \(\rm \vec{b}\) is only one.
2. The angle between the vectors is \(\frac{\pi}{3}\).
Which of the statements given above is/are correct?
Consider the following points :
1. (-1, -3, 1)
2. (-1, 3, 2)
3. (-2, 5, 3)
Which of the above points lie on the line joining A and B ?
What is the magnitude of \(\overrightarrow{A B}\) ?
If \({\rm{\vec d}} = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}\) , then which of the following equations is/are correct?
1. y – x = 4
2. 2z – 3 = 0
Select the correct answer using the code given below:
