Two independent sources of force act on an object. The vector form of these forces is given below:

$\overrightarrow{F}_I = 2 \skew{2.5}\hat{i} + 7 \skew{4}\hat{j} \ N \ $ and $ \ \overrightarrow{F}_{II} = 5 \skew{2.5}\hat{i} + 4 \skew{4}\hat{j} \ N$

What is the acceleration of the object, in $m/s^2$, if its mass is $4 \ kg$?

Look at the below Atwood Machine with a frictionless pulley. Assume the string and the pulley are of negligible mass.

The system is released from rest. The magnitude of acceleration of the $2 \ kg$ block is,

A frictionless circular shaped ring has a slidable small marble of mass $m$ attached to it, as shown below.

The marble is set into motion at a uniform speed $u$. Which of the below forces keeps the marble moving at a uniform speed?

A block of mass $2 \ kg$ is on an inclined plane. The coefficient of static friction between the block and the plane is $0.4$. How much force would be needed to set the block in motion (down the incline) if the angle of the inclined plane is $12°$?

Two blocks are connected by a massless string of length $l$. A force of $25 \ N$ acts on the right block, as shown below.

If the coefficient of friction for the right block is $0.33$, then what is the value of the coefficient of friction $μ$ for the left block?

(Assume the system remains just in equilibrium and any small force can set the system in motion.)

Two blocks are kept on each other, as shown below.

The coefficient of static friction between the two blocks is $0.25$, and the coefficient of kinetic friction is $0.18$. If a force of $40 \ N$ acts on the lower block, what will the acceleration of the upper block be?

(Assume there is no friction between the lower block and the floor.)

A particle of mass $m$ is moving through a horizontally oriented medium. The particle experiences a drag force of $γv$ in a direction opposite to the velocity. If there is no other force acting on the particle, then which of the below equations represents the motion of the particle?

A stone of mass $m$ tied to a massless string of length $r$ is set into motion. If the stone moves in a vertical circle, what is the minimum velocity that the stone must have when it is at the topmost point of the circle?

The velocity vector of a car of mass $m$ is given by the equation:

$\overrightarrow{v}(t) = 5t \skew{2.5}\hat{i} + (10 -t^2) \skew{4}\hat{j}$

What is the average force acting on the car from $t = 1$ to $t = 4$?

Questions 10, 11 and 12 are based on the below information:

Look at the below system of three blocks.

The strings and the pulley are massless and the frictional force between the blocks and the surface is zero. When the system starts from rest, the tension force between the $4 \ kg$ block and the $2 \ kg$ block is,

Look at the below system of three blocks.

What is the acceleration of the $2 \ kg$ block kept on the table, if the other $2 \ kg$ block was suddenly taken away from the system?

Look at the below system of three blocks.

At $t=t_{break}$, the string between the $4 \ kg$ block and the $2 \ kg$ block breaks. What will be the acceleration of the $2 \ kg$ block now?

A block at the bottom of an inclined plane is given a sharp push at $t = 0 \ s$.

As a result, the block moves upwards and travels a distance $h$ before it starts sliding down. The coefficient of friction between the block and the inclined plane is $0.15$, and the angle of inclination is $25°$. What is the value of $h$?

A time-dependent centripetal force acts on an object of mass $2.4 \ kg$. The expression for the centripetal force is given as,

$\overrightarrow{f}_{cent} (t)=10[\cos ⁡8t \skew{2.5}\hat{i} + \sin⁡ 8t \skew{4}\hat{j} ] N$

The object moves in a circle of radius $3 \ cm$. Find the speed of the object when $t = 6 \ s$. (Note: The angles are in radians)

The differential equation of an object moving through a viscous medium is given as:

$m \dfrac{dv}{dt} = -Av^3$

Where $A$ is a positive constant and $v(t)$ is the time-dependent velocity of the object. Find an expression for $v(t$) subject to the condition that $v(0)=1 \ m/s$.