ECAT — Engineering College Admission Test (UET)
Take the ECAT Physics — Measurements, Vectors and Equilibrium (Set 6): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. Which of the following has the same dimensions as ε₀E², where ε₀ is the permittivity of free space and E is electric field strength?
Answer: A. Energy per unit volume
Explanation: ε₀E² has units of (C²N⁻¹m⁻²)(N²C⁻²) = N m⁻² = J m⁻³, i.e. energy per unit volume (ML⁻¹T⁻²), the same as pressure.
Q2. The density of a solid cylinder is found from ρ = m/(πr²h). The measured mass, radius and height have maximum percentage uncertainties of 1%, 2% and 1% respectively. What is the maximum percentage uncertainty in ρ?
Answer: A. 6 %
Explanation: Fractional errors add, with powers as multipliers: 1 + 2×2 + 1 = 6%.
Q3. For A = 2i − j + 3k and B = i + 4j − 2k, find |A × B| correct to three significant figures.
Answer: D. 15.2
Explanation: A × B = (-10, 7, 9), so |A × B| = √230 = 15.2.
Q4. The angle θ between the vectors A = i + 2j + 2k and B = 2i − j + 2k satisfies
Answer: B. cos θ = 4/9
Explanation: A·B = 2 − 2 + 4 = 4 and |A||B| = 3 × 3 = 9, so cos θ = 4/9.
Q5. A force F = 2i − 3j + k N acts at the point with position vector r = i + 2j m from the origin. Taking τ = r × F, the torque about the origin in N m is
Answer: C. 2i − j − 7k
Explanation: r × F = (2·1 − 0·(−3))i − (1·1 − 0·2)j + (1·(−3) − 2·2)k = 2i -1j -7k, i.e. 2i − j − 7k.
Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.
Q6. A uniform ladder weighing 200 N and 10 m long leans on a smooth vertical wall with its foot on rough ground 6 m from the wall (it touches the wall 8 m up). A 600 N person stands on the ladder at a point whose horizontal distance from the foot is 4 m. What friction force does the ground exert on the ladder?
Q7. A uniform beam of mass 30 kg and length 6 m rests on two supports, one 1 m from its left end and the other at its right end. A 20 kg load hangs from the left end of the beam. Take g = 10 m/s². What is the reaction at the support nearer the left end?
Q8. A rectangle measures 3.27 cm by 1.5 cm. Using the rule that a product has as many significant figures as the least precise factor, its area in cm² is
Q9. A vernier caliper has 25 vernier divisions equal to 24 main-scale divisions, each main-scale division being 0.5 mm. With the jaws closed the 2nd vernier division coincides with a main-scale line (zero reading on the main scale), and with an object the main scale reads 7.5 mm and the 13th vernier division coincides. The corrected length in mm is
Q10. The oscillation frequency f of a small liquid drop depends on surface tension σ, density ρ and radius r as f = kσᵃρᵇrᶜ. Dimensional analysis gives the exponent c as
Q11. In terms of SI base units, the tesla is
Q12. A signal of frequency 3.6×10⁻⁴ GHz is equal to how many kilohertz?
Q13. A hiker walks 4 km north, then 3 km east, then 8 km south. How far is she from her starting point?
Q14. A vector makes angles of 60° with the x-axis and 60° with the y-axis, and an acute angle with the z-axis. What is the angle with the z-axis, in degrees?
Q15. A force F = 6i + 8j N is resolved along the line making 45° with the x-axis (unit vector (i + j)/√2). The component of F along this line is
Q16. A = 3i + j + 2k and B = i + 2j + 2k. The scalar projection of A on B is
Q17. Two sides of a triangle are represented by A = i + 2j + 2k and B = 2i + j − 2k (in cm). Its area in cm² is
Q18. Two forces F₁ = 3i + 4j N and F₂ = −7i − j N act on a particle. The magnitude of the third force that keeps the particle in equilibrium is
Q19. A 100 N weight hangs at rest from two light strings that make angles of 30° and 60° with the horizontal ceiling, on opposite sides of the weight. The tension in the string at 30° to the horizontal is
Q20. A driver applies equal and opposite tangential forces of 12 N at the two ends of a diameter of a steering wheel 0.40 m in diameter. The torque of this couple is
Q21. A uniform circular disc of radius 10 cm has a circular hole of radius 5 cm cut out, the hole's edge touching the disc's edge (so its centre is 5 cm from the disc's centre). How far, in cm, is the centre of mass of the remaining plate from the disc's centre?
Q22. Which of the following objects, when slightly displaced from rest, is in neutral equilibrium?
Q23. Resistivity is found from ρ = πd²R/(4L). The maximum percentage errors are 1% in d, 2% in R and 0.5% in L. The maximum percentage error in ρ is
Q24. Five stopwatch readings of a pendulum's period are 2.62, 2.64, 2.63, 2.65 and 2.61 s. The mean absolute deviation (average of the magnitudes of the deviations from the mean) is
Q25. A car moving at a steady 10 m/s turns through a right-angled corner without changing speed. The magnitude of the change in its velocity vector is
Q26. A screw gauge has a least count of 0.01 mm. With the studs in contact its circular scale reads 3 divisions below its zero mark (reading −3 divisions, so the zero error is −0.03 mm). A wire gives a main-scale reading of 5.00 mm and a circular-scale reading of 22 divisions. The corrected diameter in mm is
Q27. The dimensional formula ML²T⁻²A⁻¹ belongs to which physical quantity?
Q28. The time for 20 oscillations is measured as 36.4 s with a stopwatch of least count 0.1 s. What is the percentage uncertainty in the period obtained by dividing by 20?
Q29. The unit vector perpendicular to both A = i + j and B = j + k, with direction given by A × B, is
Q30. A force F = 2i + 3j + k N displaces a body by d = 4i − j + 5k m. The work done is