6 J of work is needed to push an object of mass 2 kg from point A to point B of the frictionless inclined plane. If the angle of inclination is 30°, the height of the plane is h, what is the length s of the inclined plane?
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To solve this problem, we need to calculate the height h of the inclined plane using the given angle of inclination and then use the obtained height to find the length s of the inclined plane.
Step 1: Calculate the height h of the inclined plane.
Given:
Work done (W) = 6 J
Mass of the object (m) = 2 kg
Angle of inclination (θ) = 30°
We can use the formula for work done:
W = mgh
where g is the acceleration due to gravity (approximately 9.8 m/s²).
We know that the work done is 6 J and the mass is 2 kg, so we can rearrange the formula to solve for h:
h = W / (mg)
Substituting the given values:
h = 6 J / (2 kg * 9.8 m/s²)
h ≈ 0.306 m
Step 2: Calculate the length s of the inclined plane.
In a right-angled triangle, the opposite side (h) is the height of the plane, and the hypotenuse (s) is the length of the plane.
We can use the trigonometric relationship for right-angled triangles:
sin(θ) = h / s
Rearranging the formula to solve for s:
s = h / sin(θ)
Substituting the values we have:
s = 0.306 m / sin(30°)
s ≈ 0.612 m
Therefore, the length s of the inclined plane is approximately 0.612 meters.
To find the length (s) of the inclined plane, we need to use the concept of work and energy.
We know that work is given by the formula:
Work = Force x Distance x cos(θ)
Where:
- Work is the amount of energy transferred when a force is applied over a distance.
- Force is the component of the force acting in the direction of motion.
- Distance is the magnitude of the displacement.
- θ is the angle between the force direction and the direction of motion.
In this case, we have the work (6 J) and the angle of inclination (30°). We want to find the distance (s) of the inclined plane.
The force in this case is the force required to push the object up the inclined plane.
Force = mg x sin(θ)
Where:
- m is the mass of the object (2 kg).
- g is the acceleration due to gravity (9.8 m/s²).
Substituting this force into the work formula, we have:
Work = (mg x sin(θ)) x s x cos(θ)
6 J = (2 kg x 9.8 m/s² x sin(30°)) x s x cos(30°)
Simplifying the equation:
6 J = (19.6 m/s² x 0.5) x s x (√3/2)
6 J = 9.8 m/s² x s x (√3/2)
Now, we can solve for s:
s = 6 J / (9.8 m/s² x (√3/2))
s = 6 / (9.8 x (√3/2)) m
Calculating this equation:
s ≈ 0.342 m
Therefore, the length of the inclined plane (s) is approximately 0.342 meters.
The weight is M g = 19.6 N
(Weight)* h = 6 J = work done
Solve for the elevation change, h.
That and the inclination angle will tell you the length s.
h/s = sin 30.
Get it?