A particle of mass 5 kg is being towed at a constant speed of 6 m/s on a rough horizontal plane with coefficient of friction 0.2. At a certain point the towing force is reversed in direction. Find the distance that the particle will travel before coming to rest and explain what will happen after it does so.
To find the distance that the particle will travel before coming to rest, we need to consider the forces acting on the particle.
Initially, when the particle is being towed at a constant speed of 6 m/s, the force of towing is balanced by the force of friction. The equation for the force of friction is given by:
Frictional force = coefficient of friction * normal force
In this case, the mass of the particle is 5 kg, so the weight of the particle (mg) is 5 kg * 9.8 m/s^2 = 49 N. The normal force is equal to the weight in this case since the particle is on a horizontal plane, so the normal force is also 49 N.
Therefore, the initial force of friction is:
Frictional force = 0.2 * 49 N = 9.8 N
Since the particle is moving at a constant speed, the force of towing is equal to the force of friction:
Force of towing = 9.8 N
When the towing force is reversed in direction, the direction of frictional force also reverses. The new force of friction will have the same magnitude, but in the opposite direction:
New frictional force = -9.8 N
Now, the net force acting on the particle is the difference between the force of towing and the force of friction:
Net force = Force of towing - Frictional force
Net force = 9.8 N - (-9.8 N)
Net force = 19.6 N
Since the net force is non-zero, the particle will undergo acceleration. The equation for acceleration is:
Net force = mass * acceleration
19.6 N = 5 kg * acceleration
acceleration = 19.6 N / 5 kg
acceleration = 3.92 m/s^2
Using the kinematic equation, which relates distance, initial velocity, acceleration, and time:
Distance = (initial velocity)^2 / (2 * acceleration)
Distance = (6 m/s)^2 / (2 * 3.92 m/s^2)
Distance = 3.06 m
Therefore, the particle will travel a distance of 3.06 meters before coming to rest.
After coming to rest, the particle will not undergo any further motion if no additional forces are applied. The force of friction between the particle and the rough horizontal plane prevents it from continuing to move. The particle will remain at rest until another force is applied to overcome the force of static friction.