First, we need to find the force of friction acting on the body along the inclined plane. We can do this using the equation:
friction force = coefficient of friction * normal force
where the normal force is equal to the component of the weight of the body perpendicular to the plane, given by:
normal force = mg * cos(30)
where m is the mass of the body, g is the acceleration due to gravity, and cos(30) is the cosine of the angle between the weight vector and the perpendicular to the plane.
Therefore, the friction force is:
friction force = (1/2√3) * 5kg * 9.81m/s^2 * cos(30) = 5.10N
Next, we can resolve the constant horizontal force into its component forces parallel and perpendicular to the plane. The perpendicular force is equal to:
perpendicular force = 35√3N * sin(30) = 17.5N
This force is balanced by the normal force acting on the body. Therefore, we can focus only on the parallel force, which is equal to:
parallel force = 35√3N * cos(30) = 30.4N
This force is opposed by the force of friction. Therefore, the net force acting on the body is:
net force = 30.4N - 5.10N = 25.3N
Using Newton's second law of motion, we can find the acceleration of the body along the plane:
net force = ma
where a is the acceleration of the body.
Therefore, the acceleration of the body is:
a = net force / m = 25.3N / 5kg = 5.06m/s^2
We can now use the equation for uniformly accelerated motion to find the time taken for the body to reach the top of the plane:
s = ut + 1/2at^2
where s is the displacement of the body along the plane (equal to the length of the plane), u is the initial velocity (zero), a is the acceleration, and t is the time taken.
Rearranging, we get:
t = sqrt(2s / a) = sqrt(2 * 6.3m / 5.06m/s^2) = 1.78s
Finally, we can find the speed of the body on arrival at the top of the plane using the equation:
v = u + at
where v is the final velocity, u is the initial velocity (zero), a is the acceleration, and t is the time taken.
Therefore, the speed of the body on arrival at the top is:
v = 0 + 5.06m/s^2 * 1.78s = 9.01m/s (to 3 significant figures)
A body of mass 5kg is initially at rest at the bottom of a rough inclined plane of length 6.3m. The plane is inclined at 300 to the horizontal and coefficient of friction between the body and the plane is 1/ 2√3 . A constant horizontal force of 35 √ 3N is applied to the body causing it to 3 accelerate up the plane. Find the time taken for the body to reach the top and its speed on arrival
1 answer
1 answer