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Q5.35(b):
A block of mass 15 kg is placed on a long trolley. The coefficient of static friction between the block and the trolley is 0.18. The trolley accelerates from rest with $0.5 m s^{-2}$ for 20 s and then moves with uniform velocity. Discuss the motion of the block as viewed by (b) an observer moving with the trolley.
Solution :
Step 1: Identifying the Physical System and Inertial Frame
In this problem, we analyze the motion of a mass $m = 15 \text{ kg}$ placed on a trolley. We are tasked with describing the motion from the perspective of an observer moving with the trolley. The trolley's acceleration is $a_t = 0.5 \text{ m s}^{-2}$.
The maximum force of static friction available to keep the block stationary relative to the trolley is:
$f_{s,max} = \mu_s N = \mu_s mg$ [By the definition of static friction]
$f_{s,max} = 0.18 \times 15 \text{ kg} \times 9.8 \text{ m s}^{-2} = 26.46 \text{ N}$
Step 2: Analysis of the Pseudo-Force in the Non-Inertial Frame
An observer on the trolley is in a non-inertial frame of reference because the trolley accelerates. To apply Newton’s Laws in this frame, we must introduce a pseudo-force $F_p$ acting on the block in the direction opposite to the trolley's acceleration [Per D'Alembert's Principle].
$F_p = m \times a_t$
$F_p = 15 \text{ kg} \times 0.5 \text{ m s}^{-2} = 7.5 \text{ N}$
Step 3: Evaluating Motion During Acceleration ($0 \le t \le 20 \text{ s}$)
The pseudo-force $F_p = 7.5 \text{ N}$ acts on the block. Since $F_p < f_{s,max}$ ($7.5 \text{ N} < 26.46 \text{ N}$), the force of static friction $f_s$ exactly balances the pseudo-force to keep the block at rest relative to the trolley.
- Net Force on block (in trolley frame): $\sum F = F_p - f_s = 7.5 \text{ N} - 7.5 \text{ N} = 0$
- Observation: The block remains stationary on the trolley for the duration of the acceleration phase.
Step 4: Evaluating Motion After Acceleration ($t > 20 \text{ s}$)
At $t = 20 \text{ s}$, the trolley stops accelerating ($a_t = 0$) and moves with a uniform velocity. Consequently, the pseudo-force $F_p$ becomes zero.
- Net Force on block: $0$
- Observation: Since no external or pseudo-forces act on the block, it continues to remain at rest relative to the trolley [Per Newton's First Law].
Summary of Observations
| Phase | Status of Block relative to Trolley | Reasoning |
|---|---|---|
| $0 \le t \le 20 \text{ s}$ | Stationary | Static friction ($7.5 \text{ N}$) exceeds pseudo-force ($7.5 \text{ N}$). |
| $t > 20 \text{ s}$ | Stationary | Pseudo-force vanishes; no net force acting. |
Final Solution: An observer moving with the trolley will perceive the block to be at rest at all times, as the static friction is sufficient to counteract the pseudo-force during acceleration, and no forces act once the trolley moves with uniform velocity.
More Questions from Class 11 Physics Laws of Motion Additional Exercises
- Q5.24: Figure 5.17 shows the position-time graph of a body of mass 0.04 kg. Suggest a suitable physical context for this motion. What is the time between two consecutive impulses received by the body ? What is the magnitude of each impulse ?
- Q5.25: Figure 5.18 shows a man standing stationary with respect to a horizontal conveyor belt that is accelerating with $1 m s^{-2}$. What is the net force on the man? If the coefficient of static friction between the man’s shoes and the belt is 0.2, up to what acceleration of the belt can the man continue to be stationary relative to the belt ? (Mass of the man = 65 kg.)
- Q5.26: A stone of mass m tied to the end of a string revolves in a vertical circle of radius R. The net forces at the lowest and highest points of the circle directed vertically downwards are : [Choose the correct alternative] $T_1$ and $v_1$ denote the tension and speed at the lowest point. $T_2$ and $v_2$ denote corresponding values at the highest point.
- Q5.27(a): A helicopter of mass 1000 kg rises with a vertical acceleration of $15 m s^{-2}$. The crew and the passengers weigh 300 kg. Give the magnitude and direction of the (a) force on the floor by the crew and passengers,
- Q5.27(b): A helicopter of mass 1000 kg rises with a vertical acceleration of $15 m s^{-2}$. The crew and the passengers weigh 300 kg. Give the magnitude and direction of the (b) action of the rotor of the helicopter on the surrounding air,
- Q5.27(c): A helicopter of mass 1000 kg rises with a vertical acceleration of $15 m s^{-2}$. The crew and the passengers weigh 300 kg. Give the magnitude and direction of the (c) force on the helicopter due to the surrounding air.
- Q5.28: A stream of water flowing horizontally with a speed of $15 m s^{-1}$ gushes out of a tube of cross-sectional area $10^{-2} m^2$, and hits a vertical wall nearby. What is the force exerted on the wall by the impact of water, assuming it does not rebound ?
- Q5.29(a): Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of (a) the force on the 7th coin (counted from the bottom) due to all the coins on its top,
- Q5.29(b): Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of (b) the force on the 7th coin by the eighth coin,
- Q5.29(c): Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of (c) the reaction of the 6th coin on the 7th coin.
- Q5.30: An aircraft executes a horizontal loop at a speed of 720 km/h with its wings banked at 15°. What is the radius of the loop ?
- Q5.31: A train runs along an unbanked circular track of radius 30 m at a speed of 54 km/h. The mass of the train is $10^6$ kg. What provides the centripetal force required for this purpose — The engine or the rails ? What is the angle of banking required to prevent wearing out of the rail ?
- Q5.32: A block of mass 25 kg is raised by a 50 kg man in two different ways as shown in Fig. 5.19. What is the action on the floor by the man in the two cases ? If the floor yields to a normal force of 700 N, which mode should the man adopt to lift the block without the floor yielding ?
- Q5.33(a): A monkey of mass 40 kg climbs on a rope (Fig. 5.20) which can stand a maximum tension of 600 N. In which of the following cases will the rope break: the monkey (a) climbs up with an acceleration of $6 m s^{-2}$
- Q5.33(b): A monkey of mass 40 kg climbs on a rope (Fig. 5.20) which can stand a maximum tension of 600 N. In which of the following cases will the rope break: the monkey (b) climbs down with an acceleration of $4 m s^{-2}$
- Q5.33(c): A monkey of mass 40 kg climbs on a rope (Fig. 5.20) which can stand a maximum tension of 600 N. In which of the following cases will the rope break: the monkey (c) climbs up with a uniform speed of $5 m s^{-1}$
- Q5.33(d): A monkey of mass 40 kg climbs on a rope (Fig. 5.20) which can stand a maximum tension of 600 N. In which of the following cases will the rope break: the monkey (d) falls down the rope nearly freely under gravity? (Ignore the mass of the rope).
- Q5.34(a): Two bodies A and B of masses 5 kg and 10 kg in contact with each other rest on a table against a rigid wall (Fig. 5.21). The coefficient of friction between the bodies and the table is 0.15. A force of 200 N is applied horizontally to A. What are (a) the reaction of the partition
- Q5.34(b): Two bodies A and B of masses 5 kg and 10 kg in contact with each other rest on a table against a rigid wall (Fig. 5.21). The coefficient of friction between the bodies and the table is 0.15. A force of 200 N is applied horizontally to A. What are (b) the action-reaction forces between A and B ? What happens when the wall is removed? Does the answer to (b) change, when the bodies are in motion? Ignore the difference between $\mu_s$ and $\mu_k$.
- Q5.35(a): A block of mass 15 kg is placed on a long trolley. The coefficient of static friction between the block and the trolley is 0.18. The trolley accelerates from rest with $0.5 m s^{-2}$ for 20 s and then moves with uniform velocity. Discuss the motion of the block as viewed by (a) a stationary observer on the ground,
- Q5.36: The rear side of a truck is open and a box of 40 kg mass is placed 5 m away from the open end as shown in Fig. 5.22. The coefficient of friction between the box and the surface below it is 0.15. On a straight road, the truck starts from rest and accelerates with $2 m s^{-2}$. At what distance from the starting point does the box fall off the truck? (Ignore the size of the box).
- Q5.37: A disc revolves with a speed of $33\frac{1}{3}$ rev/min, and has a radius of 15 cm. Two coins are placed at 4 cm and 14 cm away from the centre of the record. If the co-efficient of friction between the coins and the record is 0.15, which of the coins will revolve with the record ?
- Q5.38: You may have seen in a circus a motorcyclist driving in vertical loops inside a ‘death-well’ (a hollow spherical chamber with holes, so the spectators can watch from outside). Explain clearly why the motorcyclist does not drop down when he is at the uppermost point, with no support from below. What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is 25 m ?
- Q5.39: A 70 kg man stands in contact against the inner wall of a hollow cylindrical drum of radius 3 m rotating about its vertical axis with 200 rev/min. The coefficient of friction between the wall and his clothing is 0.15. What is the minimum rotational speed of the cylinder to enable the man to remain stuck to the wall (without falling) when the floor is suddenly removed ?
- Q5.40: A thin circular loop of radius R rotates about its vertical diameter with an angular frequency $\omega$. Show that a small bead on the wire loop remains at its lowermost point for $\omega \leq \sqrt{g / R}$. What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for $\omega = \sqrt{2g / R}$ ? Neglect friction.
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