A 15-kilogram block is sliding on a dry, level surface with a coefficient of dry sliding friction of 0.3. The initial speed of the block is 10/m/sec. How far does the block slide before it stops?
To find the distance the block slides before it stops, we can use the concept of work-energy principle. According to this principle, the work done on an object is equal to its change in kinetic energy.
The work done on the block can be calculated using the equation:
Work = Force * Distance
The force acting on the block is the force of friction, given by:
Force of Friction = coefficient of friction * Normal Force
The normal force can be calculated using the equation:
Normal Force = mass * gravity
Where:
- coefficient of friction is 0.3
- mass of the block is 15 kilograms
- acceleration due to gravity is 9.8 m/s^2
Let's calculate the normal force first:
Normal Force = 15 kg * 9.8 m/s^2 = 147 N
Now we can calculate the force of friction:
Force of Friction = 0.3 * 147 N = 44.1 N
The work done on the block is equal to the initial kinetic energy because it comes to a stop:
Work = (1/2) * mass * initial velocity^2
Work = (1/2) * 15 kg * (10 m/s)^2 = 750 J
Since work is equal to force times distance, we can rearrange the equation to solve for distance:
Distance = Work / Force
Distance = 750 J / 44.1 N = 17.007 m
Therefore, the block slides for approximately 17 meters before it stops.
To find how far the block slides before it stops, we can use the equations of motion along with the concept of friction.
First, let's assume that the block stops after sliding a distance "d". We need to find the value of "d".
The first step is to calculate the force of friction acting on the block. The force of friction (F) can be determined by multiplying the coefficient of friction (μ) by the normal force (N). In this case, the normal force is equal to the weight of the block, which is given by the product of the mass (m) and the acceleration due to gravity (g).
Normal force (N) = m * g
Given:
mass (m) = 15 kg
acceleration due to gravity (g) = 9.8 m/s^2
So, N = 15 kg * 9.8 m/s^2
Now, the force of friction (F) is given by:
F = μ * N
Given:
coefficient of dry sliding friction (μ) = 0.3
So, F = 0.3 * (15 kg * 9.8 m/s^2)
Next, we need to determine the force acting to oppose the motion of the block, which is the force due to kinetic friction (Fk). Since the block is moving, the force of kinetic friction is equal to the force of friction (F).
Fk = F
Now, we know that force (F) is equal to mass (m) multiplied by acceleration (a). Since the block is decelerating due to friction, acceleration (a) will have a negative value.
Fk = m * a
So, m * a = 0.3 * (15 kg * 9.8 m/s^2)
Now, we have the force of kinetic friction (Fk), which is equal to the mass (m) multiplied by the acceleration (a). We can use another equation of motion to find the value of acceleration (a).
The equation of motion is:
v^2 = u^2 + 2 * a * d
Where:
v = final velocity (0 m/s, as the block stops)
u = initial velocity (10 m/s, given)
a = acceleration (calculated above, with a negative value)
d = distance traveled (unknown)
Plugging in the known values into the equation, we can solve for "d":
(0 m/s)^2 = (10 m/s)^2 + 2 * a * d
Simplifying the equation gives us:
0 = 100 m^2/s^2 + 2 * a * d
Rearranging the equation gives us:
2 * a * d = -100 m^2/s^2
Now, substituting the value of acceleration (a) we calculated from the previous step, we have:
2 * (-0.3 * (15 kg * 9.8 m/s^2)) * d = -100 m^2/s^2
Now, we can solve for "d":
-8.82 * d = -100
Dividing both sides by -8.82 gives:
d = 100 / 8.82
Simplifying, we find:
d ≈ 11.33 meters
Therefore, the block slides approximately 11.33 meters before it stops.