A dozer pushes up a 100 kg spool of cable along a $20^ \circ$ incline road at a constant velocity as shown in the figure. The figure shows a dozer pushing a spool (diameter 400 mm) up a $20^ \circ$ incline. Point A is the contact point between the spool and the road, and Point B is the contact point between the dozer bucket and the spool. The coefficient of static friction between the dozer bucket and the spool (Point B) is 0.45, and coefficient of kinetic friction between road and the spool (Point A) is 0.15. Consider the spool only slides up the incline. The maximum normal force in N acting at Point B, is ___________ [rounded off to 1 decimal place]
1. Coordinate System and Assumptions
We define the \( x \)-axis parallel to the incline (pointing upwards) and the \( y \)-axis perpendicular to the incline. At constant velocity and with no rotation, the net force and net torque about the spool's center are zero.
Sum of forces in the \( y \)-direction (perpendicular to incline):
\[ N_A - f_B - mg \cos 20^\circ = 0 \implies N_A = mg \cos 20^\circ + f_B \]
Sum of forces in the \( x \)-direction (parallel to incline):
\[ N_B - f_A - mg \sin 20^\circ = 0 \implies N_B = mg \sin 20^\circ + f_A \]
Torque balance about the center ensures \( f_A = f_B \). Since it slides at A, \( f_A = \mu_k N_A = 0.15 N_A \).
Substituting the friction relation into the \( y \)-balance equation:
Now, finding the friction force \( f_A \):
\[ f_A = 0.15 \times 1084.52 \approx 162.68 \text{ N} \]
Finally, calculating the normal force at B (\( N_B \)):
Important Questions from Numerical Reasoning
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Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{th}$ stair. For example, $F(1) = 1$, $F(2) = 2$, $F(3) = 3$. The value of $F(5)$ is ________.- In a zoo, three lions and four tigers eat 390 kg of food every week. In another zoo, four lions and five tigers eat 500 kg of food every week. Lions and tigers eat different amounts of food, but all individuals of the same species eat the same amount. The amount of food a single lion eats per week is ________ kg.
(Answer in integer) Consider a spherical globe rotating about an axis passing through its poles. There are three points P, Q, and R situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let P, Q, and R move with speeds $v_P$, $v_Q$, and $v_R$, respectively.
Which one of the following options is CORRECT?
