An engine of mass one metric ton is ascending on an inclined plane, at an angle tan ^-1 (1/2) with horizontal, with a speed of 36km/hr. If the coefficient of friction at the surface is 1/(3^1/2) then the power is?
To calculate the power, we need to first determine the force acting on the engine and then multiply it by the velocity. Here's how you can do it step by step:
1. Find the force parallel to the incline:
The force parallel to the incline is the force that helps the engine ascend. It opposes the component of the force of gravity acting on the engine.
The component of the force of gravity acting parallel to the incline is given by:
F_parallel = m * g * sin(theta)
where:
m = mass of the engine = 1 metric ton = 1000 kg
g = acceleration due to gravity = 9.8 m/s^2
theta = angle of the incline = tan^(-1)(1/2)
Plugging in the values, we have:
F_parallel = 1000 * 9.8 * sin(tan^(-1)(1/2))
2. Find the frictional force:
The frictional force opposes the motion of the engine and acts parallel to the incline. It is given by:
F_friction = coefficient_of_friction * normal_force
The normal force is the force perpendicular to the incline. It cancels out the component of the force of gravity acting perpendicular to the incline, which is given by:
F_perpendicular = m * g * cos(theta)
The normal force can be calculated as:
normal_force = m * g * cos(theta)
Now, substitute the normal force into the frictional force equation:
F_friction = coefficient_of_friction * (m * g * cos(theta))
Plugging in the values, we have:
F_friction = 1/(3^(1/2)) * (1000 * 9.8 * cos(tan^(-1)(1/2)))
3. Calculate the net force:
The net force acting on the engine parallel to the incline is:
F_net = F_parallel - F_friction
4. Calculate the power:
Power is defined as the rate at which work is done or energy is transferred. In this case, power can be calculated as the product of force and velocity:
Power = F_net * velocity
However, we need to convert the velocity from km/hr to m/s by dividing it by 3.6.
5. Plug in the values and calculate the power.
Now, follow these steps and perform the calculations to find the power.
To find the power, we need to determine the force acting against the motion of the engine.
First, let's convert the speed of 36 km/hr to m/s:
36 km/hr = 36,000 m/3,600 s = 10 m/s
The force opposing the motion is the sum of the gravitational force (mg) and the frictional force (F_friction).
The gravitational force can be calculated as follows:
m = 1 metric ton = 1000 kg (since 1 metric ton = 1000 kg)
g = 9.8 m/s^2 (acceleration due to gravity)
Gravitational force (mg) = 1000 kg × 9.8 m/s^2 = 9800 N
The coefficient of friction (μ) is given as 1/√3.
The frictional force (F_friction) can be calculated using:
F_friction = μ × Normal force
Normal force = mg × cos(θ), where θ is the angle of the inclined plane (tan^(-1)(1/2)).
Substituting the values:
Normal force = 1000 kg × 9.8 m/s^2 × cos(tan^(-1)(1/2))
cos(tan^(-1)(1/2)) = 1 / √(1^2 + (1/2)^2) = 2 / √5
Normal force = 1000 kg × 9.8 m/s^2 × 2 / √5 = 19600 / √5 N
Substituting this value in the equation for frictional force:
F_friction = (1/√3) × (19600 / √5) N
= 19600 / (3√5) N
The total force opposing the motion is given by:
F_total = F_friction + mg
= 19600 / (3√5) N + 9800 N
= (19600 + 29400√5) / 3 N
The power (P) can be determined using the formula:
P = F_total × velocity
Substituting the values:
P = [(19600 + 29400√5) / 3 N] × 10 m/s
= (19600 + 29400√5) / 3 × 10 W
= (1960 + 2940√5) W
Therefore, the power is (1960 + 2940√5) watts.