1. The condition for a banked curve to require no friction for a car moving at speed v is based on:

A. The angle of banking, speed, and radius of curvature B. The weight of the car only C. The friction coefficient only D. The steering angle only

2. When calculating net torque on a revolving door with forces applied on opposite sides, the sign of torque depends on:

A. The mass of the forces B. The direction of rotation the forces would cause C. The distance to the axis only D. The angle of the door

3. Which of the following statements is true for the frequency \(f\) of an object undergoing SHM?

A. \(f = \frac{1}{T}\), where \(T\) is the period B. \(f = T\) C. \(f = 2\pi T\) D. \(f\) has no relation with \(T\)

4. The kinetic energy in a spring-mass system at displacement x is:

A. Maximum when x = 0 B. Maximum when x = amplitude A C. Zero at equilibrium position D. Independent of x

5. The vector nature of torque means that torque can be calculated as:

A. The scalar product of force and distance B. The cross product of position vector and force vector C. The difference between applied forces D. The sum of all forces acting tangentially

6. For a car supported on four springs, the frequency of vertical vibrations depends on:

A. The total mass supported and the combined spring constants B. The speed of the car C. The tire pressure D. The height of the springs

7. How does the choice of the pivot point affect the calculation of torques in an equilibrium problem?

A. It changes the magnitude of forces B. It changes the sum of the forces C. It changes which forces produce torque and their distances D. It does not affect the calculation

8. What component of a force produces torque about a pivot?

A. The component parallel to the position vector B. The component perpendicular to the position vector C. The total magnitude of the force D. The component directed radially inward

9. If a man applies a force at 2.00 m from the hinges at 60°, the torque exerted is \(5.20 \times 10^2\) Nm. What is the magnitude of the force?

A. 300 N B. 260 N C. 347 N D. 200 N

10. If two torques act on an object in opposite directions, one producing clockwise rotation and the other counterclockwise, their sum is:

A. Always positive B. Always zero C. The algebraic sum considering sign convention D. Always negative

11. In SHM, when the displacement is zero (object passes through the equilibrium), the potential energy is:

A. Maximum B. Zero C. Half of total energy D. Minimum but not zero

12. The force component that contributes to torque on a door when a force is applied at an angle θ is:

A. F cos θ B. F sin θ C. F tan θ D. F / sin θ

13. In uniform circular motion, the velocity of the object is related to the period T and radius A by:

A. v = 2πA / T B. v = A / T C. v = 2πT / A D. v = A × T

14. If the net torque acting on an object at rest is not zero, the object will:

A. Remain at rest B. Rotate at a constant angular velocity C. Begin rotating with increasing angular velocity D. Move linearly without rotation

15. Which condition must be met for mechanical equilibrium of an object?

A. Net external force is zero and net external torque is zero B. Net external force is zero only C. Net external torque is zero only D. Acceleration must be constant

16. If a force is directed toward or away from the axis of rotation (θ = 0° or 180°), the torque exerted is:

A. Maximum B. Zero C. Equal to the force multiplied by distance D. Negative but nonzero

17. For a door with hinges acting as the axis of rotation, a force of magnitude \(F\) applied at a distance \(r\) at an angle \(\theta\) causes torque:

A. \(\tau = r F \cos \theta\) B. \(\tau = r F \sin \theta\) C. \(\tau = r F\) D. \(\tau = F / r \sin \theta\)

18. The velocity of a mass undergoing SHM at a displacement x is:

A. Constant B. Zero at maximum displacement C. Maximum at maximum displacement D. Zero at equilibrium position

19. When torque is positive, the object tends to rotate:

A. Clockwise B. Counterclockwise C. Does not rotate D. Depends on the force direction

20. When a force acts along the line of the position vector from the axis (i.e., \(\theta = 0^\circ\)), the torque is:

A. Maximum B. Zero C. Negative maximum D. Half the maximum

21. For an object undergoing Simple Harmonic Motion (SHM), the amplitude \(A\) is:

A. The average displacement from equilibrium B. The maximum displacement from equilibrium position C. The time period of one oscillation D. The minimum displacement from equilibrium position

22. The normal force at the pivot of a plank carrying weights is equal to:

A. The sum of all downward forces B. The difference of the weights on both sides C. The net torque D. The tension in the plank

23. Torque is defined as:

A. The force applied parallel to the lever arm B. The product of the force and the distance from the pivot point, considering the angle between them C. The sum of all forces acting on a body D. The acceleration of an object in rotational motion

24. The normal force exerted by a pivot on a seesaw is found by:

A. Summing all torques about the pivot B. Applying the first condition of equilibrium for forces C. Calculating the gravitational force only D. Ignoring the mass of the plank

25. The torque \(\tau\) produced by a force \(F\) applied at a distance \(r\) from a pivot, with the force perpendicular to \(r\), is given by:

A. \(\tau = F \times r\) B. \(\tau = \frac{F}{r}\) C. \(\tau = F r \sin \theta\) D. \(\tau = F r \cos \theta\)

26. The angular frequency ω of an object on a spring is given by:

A. ω = √(k/m) B. ω = k/m C. ω = m/k D. ω = k + m

27. When a child sits at a certain position on a seesaw and another child sits on the opposite side to balance it, which quantity is used to find the unknown child's mass or position?

A. Conservation of energy B. Sum of torques about the pivot equals zero C. Sum of kinetic energies equals zero D. Conservation of linear momentum

28. In uniform circular motion, the instantaneous velocity vector is:

A. Pointing toward the center of the circle B. Pointing radially outward from the center C. Tangent to the circle D. Zero at any instant

29. The SI unit of torque is:

A. Newton (N) B. Newton meter (N·m) C. Joule (J) D. Watt (W)

30. The lever arm \(d\) in torque calculation is defined as:

A. \(d = r \cos \theta\) B. \(d = r \tan \theta\) C. \(d = r \sin \theta\) D. \(d = r\)

31. When calculating torque about the left end of a plank rather than the pivot, the gravitational force on the plank:

A. Produces zero torque B. Produces a nonzero torque C. Is ignored D. Always cancels out

32. In the force and torque notation, if torque causes clockwise rotation, the torque is considered:

A. Positive B. Negative C. Zero D. Undefined

33. In a balanced seesaw system, what condition must the sum of torques satisfy?

A. It must be a positive value B. It must be negative C. It must equal zero D. It must equal the sum of forces

34. In uniform circular motion, if the radius of the circle doubles and the velocity stays the same, the centripetal acceleration:

A. Doubles B. Halves C. Stays the same D. Quadruples

35. The velocity of an SHM object as a function of position \(x\) is expressed as:

A. \(v = \pm \sqrt{\frac{k}{m} (A^2 - x^2)}\) B. \(v = \pm \sqrt{\frac{m}{k} (A^2 - x^2)}\) C. \(v = \pm \sqrt{k m (x^2 - A^2)}\) D. \(v = \pm \frac{k}{m} (x - A)\)

36. When a force is applied at an angle \(\theta\) to the position vector \(r\), the effective torque magnitude is:

A. \(\tau = r F \cos \theta\) B. \(\tau = r F \sin \theta\) C. \(\tau = r F \tan \theta\) D. \(\tau = r F \theta\)

37. When calculating torque about a pivot point, choosing the pivot location such that one of the forces acts through it results in:

A. That force producing maximum torque B. That force producing zero torque C. That force being ignored D. The torque being infinite

38. What is the formula for the magnitude of centripetal acceleration in uniform circular motion?

A. \( a_c = \frac{v}{R} \) B. \( a_c = \frac{v^2}{R} \) C. \( a_c = vR \) D. \( a_c = \frac{R^2}{v} \)

39. The total mechanical energy \(E\) in a SHM system with spring constant \(k\) and amplitude \(A\) is:

A. \(E = \frac{1}{2} k x^2\) B. \(E = \frac{1}{2} k A^2\) C. \(E = k A\) D. \(E = k x\)

40. What is the physical significance of a net torque of zero on an object?

A. The object does not rotate B. The object rotates at a constant angular velocity C. The object accelerates angularly D. The object moves linearly

41. A rope tied 0.500 m above ground to a truck bumper 8.00 m long exerts tension to create a torque on a tree. The reference point for torque calculation is:

A. The bumper B. The base of the tree C. The tree top D. Midpoint of the rope

42. The sum of vertical forces on a seesaw system in equilibrium must satisfy:

A. The total weight of the system equals the normal force at the pivot point B. The total torque equals zero C. Only the applied forces balance out D. The normal force is zero

43. The time period \(T\) of an object in SHM with mass \(m\) and spring constant \(k\) is given by:

A. \(T = 2\pi \sqrt{\frac{m}{k}}\) B. \(T = 2\pi \sqrt{\frac{k}{m}}\) C. \(T = \frac{2\pi m}{k}\) D. \(T = \frac{2\pi k}{m}\)

44. The maximum torque exerted by a force occurs when the force is applied at what angle relative to the position vector?

A. 0° B. 45° C. 90° D. 180°

45. When choosing an axis of rotation at the pivot for a seesaw problem, the torques due to the pivot force itself are:

A. Positive B. Negative C. Zero D. Equal to the gravitational force

46. In simple harmonic motion (SHM), the period of oscillation depends on:

A. The amplitude of oscillation B. The mass and the spring constant C. The initial velocity D. The friction force

47. If a man of mass M sits on one side of a seesaw and a woman of mass m on the other side, the man’s distance x from the pivot for balance is:

A. x = (M/m)(L/2) B. x = (m/M)(L/2) C. x = (M + m)(L/2) D. x = L / (2M)

48. The force exerted by a spring is proportional to:

A. Displacement squared B. The mass attached C. Displacement linearly D. The period of motion

49. The torque caused by a tension in a rope attached to a tree is calculated with reference to:

A. The force component parallel to the tree B. The length of the rope only C. The base of the tree as the pivot point and the lever arm D. The height of the rope attachment only

50. In the seesaw problem with a woman sitting on one end and a man needing to balance at a certain point, which condition determines the balancing position of the man?

A. Sum of forces equals zero B. Sum of torques about the pivot equals zero C. Sum of kinetic energies equals zero D. Sum of potential energies equals zero