To solve this problem, we need to break it down into several parts: determining the time it takes for the body to slide down, calculating the distance covered, and finding the coefficient of sliding friction. Let's tackle each step methodically.
Understanding the Motion Up the Incline
The body is pushed up an inclined plane with an initial velocity of 14.6 m/s and reaches its maximum height in 2 seconds. At maximum height, the final velocity is 0 m/s. We can use the equations of motion to find the acceleration acting on the body.
Calculating the Acceleration
Using the formula:
Where:
- v = final velocity (0 m/s at maximum height)
- u = initial velocity (14.6 m/s)
- a = acceleration
- t = time (2 s)
Plugging in the values:
0 = 14.6 + a(2)
Solving for acceleration (a):
a = (0 - 14.6) / 2 = -7.3 m/s²
Finding the Distance Covered Up the Incline
Next, we can calculate the distance covered while moving up the incline using the formula:
Substituting the known values:
s = (14.6)(2) + (1/2)(-7.3)(2²)
s = 29.2 - 14.6 = 14.6 m
Time to Slide Down the Incline
When the body slides back down, it will start from rest at the maximum height and accelerate down the incline. The acceleration down the incline can be calculated by considering both gravity and friction.
Calculating the Forces Acting on the Body
The gravitational force acting down the incline can be calculated as:
Where:
- m = mass of the body
- g = acceleration due to gravity (approximately 9.81 m/s²)
- θ = angle of inclination (30 degrees)
The frictional force acting against the motion is:
Where μ is the coefficient of friction. The net force acting on the body when sliding down is:
- F_net = F_gravity - F_friction
Substituting the forces:
F_net = mg sin(30) - μmg cos(30)
F_net = mg(0.5) - μmg(√3/2)
F_net = mg(0.5 - μ√3/2)
Using Newton's Second Law
According to Newton's second law:
Setting the two expressions for net force equal gives:
ma = mg(0.5 - μ√3/2)
Canceling m (assuming it is not zero):
a = g(0.5 - μ√3/2)
Substituting g = 9.81 m/s²:
a = 9.81(0.5 - μ√3/2)
Finding the Coefficient of Friction
To find the coefficient of friction, we need to consider the time it takes to slide down. We can use the distance covered (14.6 m) and the kinematic equation:
For sliding down, the initial velocity (u) is 0, so:
14.6 = (1/2)(9.81(0.5 - μ√3/2))t²
Rearranging gives:
t² = (14.6 * 2) / (9.81(0.5 - μ√3/2))
t² = 29.2 / (9.81(0.5 - μ√3/2))
Now, we can find the time taken to slide down using the previously calculated acceleration and the distance. Once we have the time, we can solve for μ using the relationship derived above.
Final Steps
To summarize, we have:
- Distance covered while going up: 14.6 m
- Acceleration while sliding down: dependent on μ
- Time to slide down: can be calculated once μ is known
To find the exact value of the coefficient of friction, we would need additional information or assumptions about the mass of the body or the time taken to slide down. However, this framework provides a comprehensive approach to solving the problem using the principles of physics.
