A block of mass M = 439.0 g sits on a horizontal tabletop. The coefficients of static and kinetic friction are 0.671 and 0.441, respectively, at the contact surface between table and block. The block is pushed on with a 12.9 N external force at an angle θ with the horizontal.
a) What angle with respect to the horizontal will lead to the maximum acceleration of the block for a given pushing force?
b) What is the maximum acceleration?
To solve this problem, we need to consider the forces acting on the block and use Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.
Let's start by analyzing the forces acting on the block:
1. The gravitational force (weight): The weight of an object can be calculated using the formula W = m * g, where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s^2). In this case, W = 0.439 kg * 9.8 m/s^2.
2. The normal force (N): The normal force is the force exerted by the table on the block, perpendicular to the surface of contact. Since the block is placed on a horizontal surface and is not moving vertically, the normal force is equal in magnitude and opposite in direction to the weight of the block, so N = W.
3. The external force (F): The external force is the force applied by an external source, in this case, it is the force applied at an angle θ with respect to the horizontal. We need to decompose this force into its horizontal and vertical components.
- The horizontal component of the external force (F_cosθ): F_cosθ = F * cosθ
- The vertical component of the external force (F_sinθ): F_sinθ = F * sinθ
Now we can determine the maximum acceleration and the angle that will lead to it.
a) To find the angle with respect to the horizontal that will give the maximum acceleration, we need to consider the static friction force. When the block is at rest or just about to move, the static frictional force will oppose the external force.
The maximum static frictional force (f_s) can be calculated using the formula f_s = μ_s * N, where μ_s is the coefficient of static friction. In this case, f_s = 0.671 * W.
The maximum acceleration (a_max) occurs when the static frictional force is at its maximum value. Therefore, we need to find the angle θ such that the horizontal component of the external force (F_cosθ) equals the maximum static frictional force (f_s).
F_cosθ = f_s
F * cosθ = 0.671 * W
Substituting the known values, we can solve for θ.
b) Once we know the angle that gives the maximum acceleration, we can calculate the maximum acceleration (a_max) using the formula:
a_max = (F - f_s) / M
Substituting the known values for F, f_s, and M, we can find the maximum acceleration.