A body weighing 350 kg rests on a plane inclined 30" with the
horizontal. The angle of static friction between the body and the
plane is 15 degrees. What horizontal force P is necessary to hold
the body from sliding down the plane?
To determine the horizontal force required to hold the body from sliding down the inclined plane, we need to consider the forces acting on the body.
1. Decompose the weight of the body into its components:
- The weight acting vertically downwards can be calculated as W = mg, where m is the mass of the body (350 kg) and g is the acceleration due to gravity (9.8 m/s^2).
- The component of the weight acting parallel to the plane is W_parallel = W * sin(30°) since the plane is inclined at 30 degrees.
2. Calculate the maximum static friction force that can act on the body:
- The maximum static friction force is given by f_max = μ * N, where μ is the coefficient of static friction and N is the normal force exerted on the body by the inclined plane.
- The normal force can be calculated as N = W * cos(30°).
3. Determine if the maximum static friction force is sufficient to prevent sliding:
- If the maximum static friction force (f_max) is greater than or equal to the component of weight acting parallel to the plane (W_parallel), the body will not slide down.
4. Calculate the required horizontal force P:
- If the body is at the maximum limit of static friction, the horizontal force required to hold it in place will be equal to the component of weight parallel to the plane since the friction force balances it.
- So, P = W_parallel.
Let's calculate the values step by step:
1. Weight of the body:
W = m * g = 350 kg * 9.8 m/s^2 = 3430 N
Component of weight parallel to the plane:
W_parallel = W * sin(30°) = 3430 N * sin(30°) = 1715 N
2. Maximum static friction force:
N = W * cos(30°) = 3430 N * cos(30°) = 2971.24 N
f_max = μ * N = tan(15°) * 2971.24 N = 747.15 N
3. Compare the maximum static friction force and the component of weight parallel to the plane:
f_max >= W_parallel
747.15 N >= 1715 N
Since the maximum static friction force is not enough to prevent sliding, the body will start sliding down the plane.
4. The required horizontal force to hold the body in place is equal to the component of weight parallel to the plane:
P = W_parallel = 1715 N
Therefore, the horizontal force, P, necessary to hold the body from sliding down the plane is 1715 N.