Q.1) A uniform rod of length L and mass M is pivoted at its midpoint. It is initially at rest and then released. What will be the angular acceleration of the rod just after it is released?
- A) 2g/L
- B) 3g/L
- C) 4g/L
- D) 5g/L
Answer:
The correct answer is A) 2g/L.
Explanation:
When the rod is released, gravity exerts a torque on it, causing it to rotate about the pivot point. The torque is given by T = Iα, where I is the moment of inertia of the rod and α is the angular acceleration. Since the rod is rotating about its midpoint, the moment of inertia is (1/12)ML^2.
The torque due to gravity is Mg/2, where g is the acceleration due to gravity. Therefore, we have:
Mg/2 = (1/12)ML^2 * α
Simplifying this equation, we get:
α = 2g/L
Therefore, the angular acceleration of the rod just after it is released is 2g/L, which is option A.
Q2.) A uniform thin rod of length L and mass M is rotated about its midpoint with an angular speed of ω. If it is now rotated about one of its ends, the angular speed will be:
- A) ω
- B) 2ω
- C) 3ω/2
- D) 4ω/3
Explanation:
Initially, the rod is rotating about its midpoint with an angular speed ω. The moment of inertia of the rod about its midpoint is (1/12)ML^2. Therefore, its initial angular momentum is (1/12)ML^2 * ω.
When the rod is rotated about one of its ends, its moment of inertia changes. The moment of inertia about the end is (1/3)ML^2. Since angular momentum is conserved, we can set the initial angular momentum equal to the final angular momentum:
(1/12)ML^2 * ω = (1/3)ML^2 * ωf
Solving for ωf, we get:
ωf = (1/4)ω
Therefore, the final angular speed of the rod is (1/4)ω, which is option A.
Q3.) A rotation is a transformation that:
- a) changes the size of a figure
- b) changes the shape of a figure
- c) moves a figure around a fixed point
- d) reflects a figure across a line
Answer: c) moves a figure around a fixed point
Q4.) The fixed point of a rotation is called the:
- a) center
- b) vertex
- c) origin
- d) focus
Answer: a) center
Q5.) A rotation of 90 degrees clockwise is the same as a rotation of ____ degrees counterclockwise.
- a) 90
- b) 180
- c) 270
- d) 360
Answer: c) 270
Q6.) Which of the following is a true statement about rotations?
- a) A rotation can be performed in any direction.
- b) A rotation always changes the orientation of a figure.
- c) A rotation preserves the length of a figure's sides.
- d) A rotation changes the position of a figure's vertices.
Answer: c) A rotation preserves the length of a figure's sides.
Q7.) The image of a figure after a rotation can be found by:
- a) adding the coordinates of the original figure to the coordinates of the center of rotation
- b) subtracting the coordinates of the original figure from the coordinates of the center of rotation
- c) multiplying the coordinates of the original figure by a rotation matrix
- d) dividing the coordinates of the original figure by the angle of rotation
Answer: c) multiplying the coordinates of the original figure by a rotation matrix.
Q8.) What is rotation in geometry?
- a) Movement of a point along a line
- b) Movement of a point around a fixed point
- c) Movement of a point in a straight line
- d) Movement of a point in a circular path
Answer: b) Movement of a point around a fixed point
Q9.) What is the center of rotation?
- a) The point around which the object is rotated
- b) The point on the object that remains fixed during rotation
- c) The point on the object that moves during rotation
- d) The point from where the rotation starts
Answer: a) The point around which the object is rotated
Q10.)What is the angle of rotation?
a) The amount of movement during rotation
b) The distance covered during rotation
c) The angle through which the object is rotated
d) The speed of rotation
Answer: c) The angle through which the object is rotated
Q11.)In a clockwise rotation, what is the direction of rotation?
- a) Counter-clockwise
- b) From top to bottom
- c) From right to left
- d) From left to right
Answer: d) From left to right
Q12.) If an object is rotated 90 degrees clockwise, what will be its image?
- a) A reflection of the object
- b) An object with its sides interchanged
- c) An object rotated 90 degrees anti-clockwise
- d) An object rotated 90 degrees clockwise
Answer: d) An object rotated 90 degrees clockwise
Q13.) Which of the following is a unit of measurement for rotational speed?
- a) Watts
- b) Joules
- c) Radians
- d) RPM (Revolutions per minute)
ANS: d) RPM (Revolutions per minute)
Q14.) A point that does not move during rotation is called the:
- a) Center of mass
- b) Axis of rotation
- c) Angular momentum
- d) None of the above
ANS: b) Axis of rotation
Q15.) What is the direction of the angular velocity vector for counterclockwise rotation?
- a) Up
- b) Down
- c) Left
- d) Right
ANS: c) Left
Q16.) A rigid object is rotating with a constant angular velocity. Which of the following statements is true?
- a) The object is not experiencing any torque
- b) The object is accelerating
- c) The object is losing kinetic energy
- d) None of the above
ANS: a) The object is not experiencing any torque
Q17.) Which of the following is an example of rotational motion?
- a) A car moving along a straight road
- b) A rollercoaster going up and down hills
- c) A basketball being dribbled on a court
- d) None of the above
ANS: b) A rollercoaster going up and down hills
