Laws Of Motion

Equation of motion when the mass slides down is

Given:
Therefore, from E (1), we have
Equation of motion when mass is pushed upward is

The correct answer is (A) :
From virtual work done method,

dm = 10g
v = 4.5 cm/s
dt = 5s

i.e.

Therefore, the correct answer is option (B): 9.

From the above relation, we can derive the ratio of distances and by using the velocity ratio squared.

So, the distance ratio is:

Now, to find the distance , we multiply the ratio by :

Substituting :

The frictional force is responsible for causing the coin to slip. This force is given by , where is the angular velocity and is the distance from the center. Given that the angular velocity is halved, we use the equation . When the angular velocity is halved, the distance from the center at which the coin slips will be: Thus, when the angular velocity is halved, the coin will slip at a distance of 4 cm from the center.

Calculation of Final Velocity After Collision:

Given Data:

• Mass of first particle:
• Mass of second particle:
• Initial velocity of first particle:
• Initial velocity of second particle: (at rest)

Step 1: Initial Momentum of the System

The initial momentum is:

Step 2: Final Momentum of the System

After the collision, both particles stick together, forming a combined mass:

Let be the velocity of the combined mass. The final momentum is:

Step 3: Conservation of Momentum

Using the law of conservation of momentum:

Substitute the values:

Solve for :

Conclusion:

The final velocity of the combined mass is .

Answer: C

Step 1: Analyze the Forces Acting on Each Block The pulley system involves two blocks: on the incline. \item Block on the incline.

The forces acting on each block are: For : The tension in the string is . \item For : The tension in the string is .

Step 2: Write the Equations of Motion Since the system is in equilibrium (no acceleration), the net forces along each incline must balance out: For : For :

Step 3: Solve for the Tension Combine the equations of motion:

Final Answer The tension in the string is:

The correct answer is 3

θ = 45º
Taking components along x & y




F₁ : F₂ = 1 : 3
So , value of x is 3

Let’s analyze the forces acting on each block.

For the system as a whole (masses , , and together) moving upwards with an acceleration :

Total mass,

Total weight,

Since the entire system is accelerating upwards, the net force required to produce this acceleration is given by:

Thus, the tension in rope 1 must support both the weight and the additional force required for acceleration:

To solve the problem of determining the force exerted by the ball on the player's hand, we can use the impulse-momentum theorem. The impulse experienced by an object is equal to the change in momentum of the object. The formula we use is:

• Where is the force,
• is the time duration (0.1 s in this case), and
• is the change in momentum.

The change in momentum is given by:

• Where is the mass of the ball (120 g = 0.12 kg),
• is the final velocity (0 m/s, as the ball comes to rest), and
• is the initial velocity (25 m/s).

Substituting the values:

According to the impulse-momentum theorem:

Solve for :

The negative sign indicates the force is in the opposite direction of the initial motion, but since we are asked for the magnitude, it is 30 N. Therefore, the correct answer is:

30

To find the ratio of the masses , we start by analyzing the forces acting on each block. Consider two blocks of masses and connected by a light unstretchable string passing over a smooth pulley.

1. The block with mass experiences a gravitational force downward and a tension upward.
2. The block with mass experiences a gravitational force downward and a tension upward.
3. According to Newton's second law, for each block, the equations of motion are given as follows: (Equation 1) and (Equation 2).
4. Given that the acceleration , we can substitute it into the equations:
5. From Equation 1: .
6. From Equation 2: .
7. Equating the expressions for from both equations, we have:
   • Simplifying, we get:
   • This simplifies to:
   • Therefore, .
   • Thus, .

Hence, the correct ratio of the masses is 9:7, which matches the given correct option.

To find the mass that causes block to move down with an acceleration of , we analyze the forces acting on both blocks along the inclined planes.

Consider the forces acting on block :

• Gravitational force down the plane:
• Frictional force opposing motion:
• Tension in the string:

The net force on block is given by:

Where:

• is the coefficient of friction.
• is the acceleration.

For block on the inclined plane:

• Gravitational force down the plane:
• Frictional force opposing motion:
• Tension in the string:

The net force on block is given by:

We solve these equations simultaneously to find .

Calculations:

For block :

For block :

Thus, the mass required is 4.5 kg.

To solve this question, we need to find the velocity of a body after it has been influenced by a constant force for a specific duration.

Given:

• Initial velocity of the body:
• Mass of the body:
• Force acting on the body:
• Time duration in the force field:

To find the final velocity, we can use the equation of motion under constant acceleration:

First, let's calculate the acceleration using Newton's second law:

Substitute the values into the velocity equation:

Further simplify the equation:

Thus, the velocity of the body when it emerges from the force field is: .

Hence, the correct answer is:

We are given the following information:

• Radius of curvature of the convex mirror:
• Speed of the approaching car:
• Distance of the car from the mirror: (negative because the car is approaching the mirror)

The formula for the magnification of a convex mirror is given by:

Where:

• focal length
• Image distance

Using the formula for the acceleration of the image , we can find the acceleration of the image. After applying necessary steps and simplifications, we find:

Therefore, the value of is approximately:

8 m/s²

Answer:

The value of is 8 m/s².

Concept: For motion starting from rest with constant acceleration, If the distance is the same in two cases, then Acceleration on a smooth incline: Acceleration on a rough incline: Step 1: Write the relation between the times.} Time on rough surface is more than smooth surface. Step 2: Use equal distance condition.} Step 3: Substitute accelerations.} Cancel : Step 4: Substitute .}