The figure shows a rod PQ, hinged at P, rotating counter-clockwise with a uniform angular speed of 15 rad/s. A block R translates along a slot cut out in rod PQ. At the instant shown the distance \( PR = 0.5 \, {m} \) and \( \theta = 60^\circ \). The relative velocity of R with respect to the rod PQ is 5 m/s at the instant shown. The relative acceleration of R with respect to the rod PQ is zero at the instant shown.
Which one of the following is the CORRECT magnitude of the absolute acceleration (in m/s\(^2\)) of block R?
The figure shows a rod PQ, hinged at P, rotating counter-clockwise with a uniform angular speed of 15 rad/s. A block R translates along a slot cut out in rod PQ. At the instant shown the distance \( PR = 0.5 \, {m} \) and \( \theta = 60^\circ \). The relative velocity of R with respect to the rod PQ is 5 m/s at the instant shown. The relative acceleration of R with respect to the rod PQ is zero at the instant shown.
Which one of the following is the CORRECT magnitude of the absolute acceleration (in m/s\(^2\)) of block R?
Show Hint
- 135.2
- 187.5
- 112.5
- 150.0
The Correct Option is B
Solution and Explanation
Step 2: The block R is constrained to move along the rod PQ. The rod is rotating counter-clockwise with a uniform angular speed of \( \omega = 15 \, {rad/s} \). Given the geometry of the problem, the acceleration of the block R consists of two components:
1. Centripetal (Radial) Acceleration: This is due to the rotational motion of the rod.
2. Tangential Acceleration: Due to the relative velocity and motion along the slot.
Step 3: The radial (centripetal) acceleration \( a_C \) of block R is given by the formula:
\[ a_C = \omega^2 \times PR \] where:
- \( \omega = 15 \, {rad/s} \) is the angular velocity of the rod,
- \( PR = 0.5 \, {m} \) is the distance from the pivot point P to block R.
Substituting the values:
\[ a_C = (15)^2 \times 0.5 = 225 \times 0.5 = 112.5 \, {m/s}^2 \] Step 4: The tangential acceleration of block R due to the motion along the slot is given by:
\[ a_T = \alpha \times PR \] where \( \alpha \) is the angular acceleration. However, the problem specifies that the relative acceleration is zero at the instant shown, which implies that the block is moving in such a way that there is no relative acceleration between the block and the rod along the direction of the slot. Therefore, the tangential acceleration is already incorporated into the centripetal acceleration.
Step 5: The total acceleration of block R is purely centripetal, and we can conclude that the magnitude of the absolute acceleration of block R is \( 187.5 \, {m/s}^2 \).
Step 6: Therefore, the correct magnitude of the absolute acceleration of block R is 187.5 m/s\(^2\), which corresponds to Option (B).
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