Laws Of Motion

To solve the problem, we need to determine the total distance travelled by a ball before it hits the plane for the second time after falling freely from a height onto a rigid horizontal surface, given the coefficient of restitution .

1. Understanding the Motion:
- When the ball is dropped from a height , it travels a distance downward.
- It then rebounds. The height it reaches after the first bounce is (since and height is proportional to square of velocity).
- After reaching this new height, it falls back down covering the same distance .

2. Total Distance Travelled:
Total distance before hitting the ground second time =
Descent (initial fall) + Ascent (after first bounce) + Descent (after reaching max height)

Final Answer:
The total distance travelled by the ball before hitting the plane the second time is .

To solve the problem, we need to find the maximum height reached by a ball thrown vertically upward with the same velocity that gives a maximum horizontal range of 80 m when thrown at an optimal angle.

1. Understanding the Projectile Range Formula:
The maximum horizontal range of a projectile (when projected at an angle of 45°) is given by:

where:
= range (80 m),
= initial velocity,
= acceleration due to gravity ( ).

2. Rearranging the Formula to Find :
We rearrange the equation to solve for :

3. Using Vertical Motion Formula to Find Maximum Height:
When the ball is thrown vertically with the same velocity, the maximum height is given by:

Substitute the value of :

Final Answer:
The maximum height reached by the ball is 40 meters.

To solve the problem, we need to determine the apparent weight of a man in a lift that is accelerating downward with an acceleration equal to gravity.

1. Understanding Apparent Weight:
Apparent weight is the normal force exerted by the floor on the person. It is given by the equation:

where:
= mass of the man = 50 kg,
= acceleration due to gravity = 9.8 m/s²,
= acceleration of the lift = 9.8 m/s² (since it's moving down with acceleration equal to gravity).

2. Substituting the Values:

3. Interpretation:
When the lift moves downward with acceleration equal to gravity, the person experiences weightlessness because the normal force becomes zero. This is similar to free fall.

Final Answer:
The apparent weight of the man is 0 N.

To solve the problem, we need to determine the apparent weight of a man standing on a weighing machine inside a lift that is falling freely under gravity.

1. Understanding Apparent Weight in Free Fall:
- Actual weight of the man is given by:

- But apparent weight is the normal force (reading on the weighing machine).
- In free fall, the acceleration of the lift is equal to , and hence there is no contact force acting upward on the man.
- Therefore, the man experiences weightlessness.

2. Apparent Weight Calculation:
Apparent weight
Since the lift is falling with acceleration :

Final Answer:
The weight shown on the weighing machine is 0 N (i.e., the man experiences weightlessness).

Step 1: Understanding the Concept of Center of Mass
- The spheres are made of the same material, meaning their masses are proportional to their volumes. - The mass of a sphere is given by: where is the density of the material.

Step 2: Center of Mass Formula for Two Particles
- The center of mass for two objects is given by:

Step 3: Substituting Masses of Spheres
Since mass is proportional to :

Step 4: Conclusion
Since the distance of the center of mass from the point of contact is , Option (4) is correct.

The center of mass of the 'a' is the combined center of mass of two parts:
1. The horizontal plate with length and width .
2. The vertical plate with length and width . The total mass of each plate is proportional to its area. Let the masses of the horizontal and vertical plates be and , respectively. When the alphabet ‘T’ is placed on the horizontal surface, the center of mass is at a certain position. Upon inverting the alphabet, the center of mass shifts accordingly. The shift in the position of the center of mass of the inverted â is calculated by finding the difference between the center of mass of the horizontal plate and the vertical plate. Since the plates are uniform, the center of mass of the horizontal plate is at from one edge, and for the vertical plate, the center of mass is at . After calculating, we find that the shift in the position of the center of mass from the horizontal surface is .

Step 1: Understanding the SHM Equation
- The given displacement equation follows: - The amplitude of oscillation is .

Step 2: Finding the Distance Traveled from to
- At : - At : - The particle moves from (extreme) → 0 → (extreme) → 0 (mean position). - Total distance traveled = .

Step 3: Finding the Final Position
- At , , which means the particle is at the mean position.

Step 4: Conclusion
Since the total distance traveled is and the final position is the mean position, Option (3) is correct.

Step 1: Understanding the Range of Gravitational Force Gravitational force is a fundamental force of nature that acts between any two masses. Unlike other fundamental forces (such as weak nuclear force or strong nuclear force), gravity: - Has an infinite range. - Gets weaker with increasing distance but never fully disappears.

Step 2: Explanation The gravitational force between two masses and is given by Newton's Law of Universal Gravitation: where is the gravitational constant and is the distance between the two masses. As increases, decreases, but it never becomes exactly zero. This indicates that gravitational force has an infinite range.

Thus, the correct answer is:

Step 1: Understanding the Relationship The acceleration due to gravity in a simple pendulum is given by: Taking the percentage error on both sides:

Step 2: Substituting Given Values Given: Thus, the error in the estimation of acceleration due to gravity is:

The equation of the wave is given by , where m, is the wave number, and is the angular frequency. The general relation for the wave number is , and the relation for the angular frequency is , where is the wavelength and is the frequency.

Step 1: From the given equation, we have and . Now, using the relation , we can find :

Step 2: Next, we use the relation , where is the speed of the wave. We can find the speed using the angular frequency relation : Now, the speed is