A body of mass 2kg, moving at 3ms-¹ on a rough horizontal floor is brought to rest after sliding through a distance of 2.5m on the floor. Calculate the coefficient of sliding friction (g=10ms-²).
Please I need the full solution.
energy (kinetic) is ... 1/2 * 2 * 3^2
work done by friction equals initial energy
frictional force * distance = K.E.
frictional force = weight * coefficient
m g * μ * 2.5 = K.E.
solve for μ
To find the coefficient of sliding friction, we can use the equation for kinetic friction:
frictional force = coefficient of sliding friction * normal force
First, let's find the normal force acting on the body. The normal force is equal to the weight of the body, which is given by:
weight = mass * acceleration due to gravity
weight = 2 kg * 10 m/s² = 20 N
Since the body is at rest, the frictional force acting on it is equal in magnitude and opposite in direction to the force applied by friction. Therefore, we can equate the frictional force to the force of friction:
frictional force = force of friction
The frictional force is given by the equation:
frictional force = mass * acceleration
frictional force = 2 kg * acceleration
The acceleration can be calculated using the equation of motion:
final velocity² = initial velocity² + 2 * acceleration * distance
Since the body is brought to rest, the final velocity is 0 m/s. The initial velocity is given as 3 m/s, and the distance traveled is 2.5 m. Substituting these values into the equation, we have:
(0 m/s)² = (3 m/s)² + 2 * acceleration * 2.5 m
0 m²/s² = 9 m²/s² + 5 m/s² * acceleration
Rearranging the equation and solving for acceleration:
-9 m²/s² = 5 m/s² * acceleration
acceleration = -9 m²/s² / (5 m/s²)
acceleration = -1.8 m/s²
Now we know the acceleration, we can substitute this value back into the equation for frictional force:
frictional force = 2 kg * (-1.8 m/s²)
frictional force = -3.6 N
Since the frictional force is opposing motion, its value should be negative.
Now, we can solve for the coefficient of sliding friction by rearranging the equation:
frictional force = coefficient of sliding friction * normal force
-3.6 N = coefficient of sliding friction * 20 N
Coefficient of sliding friction = -3.6 N / 20 N
Coefficient of sliding friction = -0.18
However, the coefficient of sliding friction cannot be negative, so we take the magnitude of the value:
Coefficient of sliding friction = 0.18
Therefore, the coefficient of sliding friction is 0.18.
To find the coefficient of sliding friction, we need to use the formula:
frictional force = coefficient of sliding friction × normal force
Here is the step-by-step solution:
1. Determine the initial kinetic energy of the body:
The initial kinetic energy (K₁) is given by the formula: K₁ = 0.5 × mass × velocity²
Substituting the given values: K₁ = 0.5 × 2 kg × (3 m/s)² = 9 J
2. Determine the work done by the frictional force:
The work done (W) is given by the formula: W = frictional force × distance
Since the body comes to rest, the work done by the frictional force is equal to the initial kinetic energy.
Therefore, W = K₁ = 9 J
3. Determine the frictional force:
The frictional force (F) is given by the formula: F = mass × acceleration
Since the body comes to rest, the acceleration is the deceleration caused by the frictional force.
The distance traveled on the floor can be used to calculate the acceleration:
Distance = initial velocity × time + 0.5 × acceleration × time²
Plugging in the given values: 2.5 m = 3 m/s × t + 0.5 × a × t²
Rearranging the equation and using the kinematic equation v = u + at (where the final velocity is 0): a = -(v² - u²) / (2d)
Thus, a = -(0 - (3 m/s)²) / (2 × 2.5 m) = -9/5 m/s²
Now, calculating the frictional force: F = 2 kg × (-9/5 m/s²) = -18/5 N
(The negative sign shows that the force acts in the opposite direction of motion.)
4. Determine the normal force:
The normal force (N) is equal to the weight of the body, which is given by the formula: weight = mass × acceleration due to gravity (g)
Therefore, N = 2 kg × 10 m/s² = 20 N
5. Determine the coefficient of sliding friction:
We can substitute the values we obtained into the formula: F = coefficient of sliding friction × N
-18/5 N = coefficient of sliding friction × 20 N
Solving for the coefficient of sliding friction:
coefficient of sliding friction = (-18/5 N) / (20 N) = -0.9
However, the coefficient of friction cannot be negative in reality. So, taking the absolute value:
coefficient of sliding friction = 0.9
Therefore, the coefficient of sliding friction for this scenario is 0.9.