Dow University MDCAT / Entry Test Physics Vectors & Equilibrium — Set 3

Dow University MDCAT / Entry Test Physics Vectors & Equilibrium — Set 3

1. Question 1

   Q1. A student pulls a block with a 6 N force eastward and an 8 N force northward. What is the magnitude of the resultant force?

   • A) 14 N
   • B) 10 N
   • C) 2 N
   • D) 12 N

   Answer: 10 N

   Explanation: The formula R = sqrt(A^2 + B^2) applies when vectors are at 90 degrees. Option A fails because it ignores the vector direction.

2. Question 2

   Q2. In a three-dimensional Cartesian coordinate system, what is the result of the scalar product between the unit vectors i and j?

   • A) 1
   • B) k
   • C) 0
   • D) -1

   Answer: 0

   Explanation: The dot product of perpendicular unit vectors is zero because cos(90) is zero. Option B is incorrect as it applies to parallel vectors.

3. Question 3

   Q3. A mechanic uses a wrench to loosen a nut. Which formula correctly represents the magnitude of the torque applied at an angle θ?

   • A) τ = r F sinθ
   • B) τ = r F cosθ
   • C) τ = r F tanθ
   • D) τ = r / F

   Answer: τ = r F sinθ

   Explanation: Torque is the product of force and the perpendicular distance (l = r sinθ). Option B fails because it uses the parallel component.

4. Question 4

   Q4. A force of 10 N acts at an angle of 30 degrees with the positive x-axis. What is the magnitude of its horizontal component?

   • A) 10 sin(30) N
   • B) 10 sin(60) N
   • C) 10 tan(30) N
   • D) 10 cos(30) N

   Answer: 10 cos(30) N

   Explanation: The horizontal component is found using A cosθ. Option B is incorrect because it represents the vertical component in standard orientation.

5. Question 5

   Q5. When calculating the vector product of two vectors A and B, what is the orientation of the resulting vector relative to A and B?

   • A) Parallel to both vectors
   • B) Perpendicular to the plane of A and B
   • C) At a 45 degree angle to vector A
   • D) In the same direction as vector B

   Answer: Perpendicular to the plane of A and B

   Explanation: The cross product of two vectors is always perpendicular to the plane containing them. Option A fails because it describes the dot product.

6. Question 6

   Q6. Two vectors are oriented in the same direction such that the angle between them is zero. What is the magnitude of their cross product?

   • A) AB
   • B) 1
   • C) 0
   • D) -AB

   Answer: 0

   Explanation: The cross product magnitude is AB sinθ; since sin(0) is zero, the product is zero. Option A describes the dot product result.

7. Question 7

   Q7. A particle is located at a point P with coordinates (3, 4) in the xy-plane. What is the correct expression for its position vector?

   • A) r = 3i + 4j
   • B) r = 4i + 3j
   • C) r = 7ij
   • D) r = 5(i + j)

   Answer: r = 3i + 4j

   Explanation: The position vector is defined as r = xi + yj. Option B is incorrect because it swaps the unit vectors for the coordinates.

8. Question 8

   Q8. A heavy crate rests motionless on a horizontal floor. Which mathematical condition must be satisfied for the translational equilibrium of the crate?

   • A) ΣF > 0
   • B) ΣF = 0
   • C) Στ ≠ 0
   • D) Fx = Fy

   Answer: ΣF = 0

   Explanation: Static equilibrium requires the vector sum of all forces to be zero. Option D is incorrect because it only considers one dimension.

9. Question 9

   Q9. Given two vectors A = i + j and B = 3i + 4j, what is the approximate angle between them calculated using the dot product formula?

   • A) 90 degrees
   • B) 0 degrees
   • C) 60 degrees
   • D) 45 degrees

   Answer: 45 degrees

   Explanation: Using cosθ = (A.B)/(|A||B|), (3+4+0)/(sqrt(26)*sqrt(2)) leads to 45 degrees. Option A is a common guess for perpendicular vectors.

10. Question 10

    Q10. A 50 cm long rod is pivoted at one end. A force of 10 N is applied at the other end at 30 degrees. What is the torque?

    • A) 0.5 Nm
    • B) 5.0 Nm
    • C) 2.5 Nm
    • D) 4.3 Nm

    Answer: 2.5 Nm

    Explanation: Torque is r multiplied by the perpendicular component of F. Option D fails because it uses the wrong trigonometric function for the lever arm.

11. Question 11

    Q11. If the magnitude of the resultant of two equal vectors is equal to the magnitude of either vector, what is the angle between them?

    • A) 90 degrees
    • B) 120 degrees
    • C) 60 degrees
    • D) 180 degrees

    Answer: 120 degrees

    Explanation: If R=A=B, the law of cosines R^2 = A^2 + B^2 + 2AB cosθ requires cosθ = -1/2. Option A is for perpendicular vectors.

12. Question 12

    Q12. At what specific angle between two non-zero vectors will the magnitude of their scalar product be equal to the magnitude of their vector product?

    • A) 45 degrees
    • B) 90 degrees
    • C) 0 degrees
    • D) 180 degrees

    Answer: 45 degrees

    Explanation: AB cosθ = AB sinθ implies tanθ = 1, so θ = 45 degrees. Option B is where the cross product is maximum.

13. Question 13

    Q13. Three vectors of equal magnitude A are joined head-to-tail to form an equilateral triangle. What is the magnitude of their resultant vector?

    • A) 3A
    • B) A sqrt(3)
    • C) 0
    • D) A

    Answer: 0

    Explanation: If vectors form a closed loop, the start and end points coincide, making the resultant zero. Option A assumes they add linearly.

14. Question 14

    Q14. A force F = 3i + 4j Newtons displaces a body by d = 2i + 3j meters. What is the work done by the force?

    • A) 12 J
    • B) 7 J
    • C) 24 J
    • D) 18 J

    Answer: 18 J

    Explanation: Work is the dot product F.d = (3*2) + (4*3) = 18. Option B is incorrect because it adds components without multiplying.

15. Question 15

    Q15. A uniform plank is balanced on a sharp pivot at its center. Which condition specifically ensures the plank does not rotate when weights are added?

    • A) ΣF = 0
    • B) Στ = 0
    • C) ΣFx = 0
    • D) Σp = 0

    Answer: Στ = 0

    Explanation: Second condition of equilibrium (Στ = 0) prevents rotation. Option A only ensures the object does not move linearly.

16. Question 16

    Q16. What is the unit vector in the direction of the vector A = 3i + 4j?

    • A) (3i + 4j) / 5
    • B) (3i + 4j) / 7
    • C) (3i + 4j) / 25
    • D) (3i + 4j) / 1

    Answer: (3i + 4j) / 5

    Explanation: A unit vector is the vector divided by its magnitude (sqrt(3^2 + 4^2) = 5). Option C incorrectly uses the square of the magnitude.

17. Question 17

    Q17. Two vectors have magnitudes of 10 units and 5 units. What is the ratio of the maximum possible resultant to the minimum possible resultant?

    • A) 1.5
    • B) 2.0
    • C) 3.0
    • D) 5.0

    Answer: 3.0

    Explanation: Rmax = A+B and Rmin = A-B. (A+B)/(A-B) = 15/5 = 3. Option A ignores the subtraction in the denominator.

18. Question 18

    Q18. Which mathematical vector operation correctly defines the torque vector τ in terms of position vector r and force vector F?

    • A) F x r
    • B) r x F
    • C) r . F
    • D) r + F

    Answer: r x F

    Explanation: Torque is defined as r x F. F x r results in the opposite direction. Option C is a scalar and cannot represent torque.

19. Question 19

    Q19. A block of weight W is held in equilibrium on a frictionless incline of angle θ by a string. What is the tension?

    • A) W cosθ
    • B) W tanθ
    • C) W / sinθ
    • D) W sinθ

    Answer: W sinθ

    Explanation: The component of weight acting down the plane is W sinθ. Option A is the normal force component.

20. Question 20

    Q20. If the magnitude of the sum of two vectors is equal to the magnitude of their difference, what is the angle between the vectors?

    • A) 90 degrees
    • B) 0 degrees
    • C) 180 degrees
    • D) 45 degrees

    Answer: 90 degrees

    Explanation: If |A+B| = |A-B|, then the vectors must be perpendicular (90 degrees). Option B results in |A+B| being maximum.