a rough 4m plank is used as 500N bag up to the floor of a lorry 2m high . 200j of work is done against friction between the bag and the plank calculate the angle of inclination of the plank to the horizontal plank
To calculate the angle of inclination of the plank to the horizontal, we can use the formula:
sin(theta) = height / hypotenuse
where theta is the angle of inclination, height is the height of the lorry (2m), and hypotenuse is the length of the plank (4m).
First, let's calculate the hypotenuse using the Pythagorean theorem:
hypotenuse = sqrt((height)^2 + (plank length)^2)
= sqrt((2m)^2 + (4m)^2)
= sqrt(4m^2 + 16m^2)
= sqrt(20m^2)
= 4.47m (rounded to 2 decimal places)
Next, let's calculate the sin(theta):
sin(theta) = height / hypotenuse
= 2m / 4.47m
= 0.447
To find the angle theta, we can take the inverse sine (arcsin) of 0.447:
theta = arcsin(0.447)
= 26.55 degrees (rounded to 2 decimal places)
Therefore, the angle of inclination of the plank to the horizontal is approximately 26.55 degrees.
To find the angle of inclination of the plank to the horizontal, we can use the concept of work and energy.
Given:
Mass of the bag = 500 N
Height of the lorry = 2 m
Work done against friction = 200 J
First, let's find the gravitational potential energy (GPE) gained by the bag as it is lifted to the height of the lorry. The GPE is given by the formula:
GPE = m * g * h
where,
m = mass of the bag (500 N)
g = acceleration due to gravity (9.8 m/s^2)
h = height (2 m)
GPE = 500 * 9.8 * 2
GPE = 9800 J
Since the work done against friction is equal to 200 J, we can find the net work done on the bag by subtracting the work against friction from the GPE:
Net Work = GPE - Work against friction
Net Work = 9800 - 200
Net Work = 9600 J
Now, let's find the work done by the plank on the bag. The work done is given by the formula:
Work = force * displacement * cos(theta)
where,
force = weight of the bag (500 N)
displacement = distance moved by the bag on the inclined plank (4 m)
theta = angle of inclination of the plank to the horizontal
Work = 500 * 4 * cos(theta)
Since we know that the net work done is 9600 J and the work done by the plank is 500 * 4 * cos(theta), we can set up the equation:
9600 = 500 * 4 * cos(theta)
Simplifying the equation:
9600 = 2000 * cos(theta)
cos(theta) = 9600 / 2000
cos(theta) = 4.8
However, the value of cos(theta) should be less than or equal to 1. Since 4.8 is greater than 1, it is not a valid value.
Therefore, there is no angle of inclination of the plank to the horizontal that satisfies the given conditions.
To calculate the angle of inclination of the plank, we can use the concept of work done. We know that work done is equal to the force applied multiplied by the distance moved in the direction of the force. In this case, the work is done against the friction between the bag and the plank.
Let's break down the problem step by step:
1. Determine the work done against friction:
Work = Force x Distance
Work = 200 J
2. Calculate the force applied:
Force = Work / Distance
Force = 200 J / 4 m
Force = 50 N
3. Calculate the vertical component of the force:
The vertical component of the force is the weight of the bag, which is equal to 500 N.
Vertical Component = 500 N
4. Determine the angle of inclination:
The angle of inclination can be calculated using trigonometry. In this case, we want to find the angle between the weight vector and the horizontal plank.
tan(angle) = Vertical Component / Force
tan(angle) = 500 N / 50 N
tan(angle) = 10
Taking the arctan of both sides gives us:
angle = arctan(10)
Using a calculator:
angle ≈ 84.3 degrees
Therefore, the angle of inclination of the plank to the horizontal is approximately 84.3 degrees.