An object of mass 2 kg is attached to a string to form a pendulum. It is initially displaced to a height of 10 cm above the lowest point and let go. It hits another object of mass 1 kg at its lowest point. Given that the collision is elastic, find the distance traveled by the 1-kg object before it comes to rest due to the friction of the surface, which is constant and has a magnitude of 5 N.

To find the distance traveled by the 1-kg object before it comes to rest due to friction, we need to consider the conservation of mechanical energy and the work done by friction.

Step 1: Determine the potential energy of the 2-kg object at its initial displacement of 10 cm above the lowest point.
The potential energy of an object at a height h is given by the formula PE = mgh, where m is the mass (2 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (10 cm = 0.1 m).

PE = (2 kg) * (9.8 m/s^2) * (0.1 m)
PE = 1.96 J

Step 2: Determine the speed of the 2-kg object just before collision.
The potential energy of the 2-kg object at its initial height is converted into kinetic energy just before the collision. Therefore, we can equate the potential energy to the kinetic energy.

KE = PE
1/2 * m * v^2 = 1.96 J

simplifying,
v^2 = (2 * 1.96 J) / (2 kg)
v^2 = 1.96 m^2/s^2
v = √(1.96) m/s
v ≈ 1.4 m/s

Step 3: Determine the work done by the friction.
The work done by friction can be calculated using the formula W = F * d, where W is the work done, F is the force of friction (5 N), and d is the distance traveled by the 1-kg object.

W = F * d
5 N * d = W

Step 4: Determine the kinetic energy of the 1-kg object just after collision.
Since the collision is elastic, the total mechanical energy is conserved. Therefore, the kinetic energy of the 2-kg object just before the collision is equal to the kinetic energy of the 1-kg object just after the collision.

KE2 = KE1
1/2 * m1 * v1^2 = 1/2 * m2 * v2^2

Substituting the known values,
1/2 * (1 kg) * 0^2 = 1/2 * (1 kg) * v2^2
0 = 1/2 * v2^2
v2 = 0 m/s

Step 5: Solve for the distance traveled by the 1-kg object before it comes to rest.
Since the kinetic energy of the 1-kg object just after the collision is zero, all the initial kinetic energy is converted into work done by friction.

W = (1/2) * m2 * v1^2
5 N * d = (1/2) * (1 kg) * (1.4 m/s)^2
d = [(1/2) * (1 kg) * (1.4 m/s)^2] / 5 N
d = 0.196 m

Therefore, the distance traveled by the 1-kg object before it comes to rest due to friction is approximately 0.196 meters.

To find the distance traveled by the 1-kg object before it comes to rest, we need to analyze the energies involved in the system.

Let's first calculate the potential energy of the 2-kg object when it's at a height of 10 cm above the lowest point. The potential energy equation is given by:

Potential Energy = mass * gravitational acceleration * height

Potential Energy = 2 kg * 9.8 m/s^2 * 0.1 m
Potential Energy = 1.96 J

Since the collision between the two objects is elastic, the total mechanical energy of the system remains conserved. Therefore, the potential energy of the 2-kg object is converted into the kinetic energy of the 1-kg object.

The equation for kinetic energy is given by:

Kinetic Energy = (1/2) * mass * velocity^2

Since the 1-kg object comes to rest due to friction, we know that its final kinetic energy is zero. Therefore, the initial kinetic energy of the 1-kg object is equal to the potential energy of the 2-kg object.

So, we have:

Kinetic Energy = 1.96 J

Now, let's use the equation for kinetic energy to find the initial velocity of the 1-kg object:

1.96 J = (1/2) * 1 kg * velocity^2
3.92 J = velocity^2

Taking the square root of both sides, we get:

velocity ≈ 1.98 m/s

Now, we can find the distance traveled by the 1-kg object before it comes to rest. To do this, we'll calculate the work done against friction.

The work done against friction is given by the equation:

Work = force * distance

We know the force of friction is 5 N. Let's assume the distance traveled by the 1-kg object before coming to rest is d. Therefore, the work done against friction is equal to the change in kinetic energy.

Work = 5 N * d

Since the initial kinetic energy is 1.96 J and the final kinetic energy is 0, the change in kinetic energy is:

Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
Change in Kinetic Energy = 0 J - 1.96 J
Change in Kinetic Energy = -1.96 J

According to the work-energy theorem, the work done against friction is equal to the change in kinetic energy.

Work = -1.96 J

But work is defined as force multiplied by distance. So, we can rewrite the equation as:

force * distance = -1.96 J

Substituting the value of the force (5 N) into the equation, we get:

5 N * distance = -1.96 J

Solving for distance, we find:

distance = -1.96 J / 5 N
distance ≈ -0.392 m

The distance traveled by the 1-kg object before it comes to rest is approximately 0.392 meters. However, it's important to note that distance cannot be negative in this context. Therefore, the object comes to rest after traveling a distance of approximately 0.392 meters.