What horizontal force is needed to push a hospital bed of mass 120 kg at a constant speed up a smooth sloping plane inclined at an angle of 37 degrees above the ground?
m*g = 120kg * 9.8N/kg = 1176 N. = Wt. of
the bed.
Fp = 1176*sin37 = 707.73 N. = Force
parallel to the incline.
Fn = 1176*cos37 = 939.20 N. = Normal =
Force perpendicular to the incline.
Fap-Fp = m*a
Fap-707.73 = m*0 = 0
Fap = 707.73 N. = Force applied parallel
to the incline.
Fx = Fap*cos A = 707.73*cos37 = 565 N.
= Hor. force applied.
To calculate the horizontal force needed to push the hospital bed up the inclined plane at a constant speed, we need to consider the forces acting on the bed.
First, let's break down the forces:
1. Weight (W): This is the force exerted by gravity on the bed and can be calculated using the formula W = m * g, where m is the mass of the bed (120 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2). In this case, W = 120 kg * 9.8 m/s^2 = 1176 N.
2. Normal Force (N): This is the perpendicular force exerted by the inclined plane on the bed. It acts perpendicular to the surface of the plane. On a smooth inclined plane, the normal force is equal in magnitude to the component of the weight of the object perpendicular to the plane. Therefore, N = W * cos(θ), where θ is the angle of inclination (37 degrees). N = 1176 N * cos(37) = 938.4 N.
3. Frictional Force (Ff): This is the force opposing the motion of the bed. Since the bed is moving at a constant speed, the frictional force would be equal to the force applied. We denote this force by F.
Now, the equation for the horizontal forces is as follows:
F - Ff = 0
Since the bed is moving at a constant speed, the frictional force Ff can be calculated using the equation:
Ff = μ * N
where μ is the coefficient of friction between the bed and the inclined plane. However, since the problem states that the inclined plane is "smooth," it means that there is negligible friction acting on the bed (μ = 0). Therefore, Ff = 0.
From the equation F - Ff = 0, we can conclude that F = 0. Therefore, no horizontal force is needed to push the hospital bed up the smooth inclined plane at a constant speed. The weight of the bed is balanced by the normal force exerted by the plane.