A student, crazed by final exams, uses a force of magnitude 86 N and angle θ = 61° to push a 5.2 kg block across the ceiling of his room, as in Figure 6-52. If the coefficient of kinetic friction between the block and the ceiling is 0.34, what is the magnitude of the block's acceleration?
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To find the magnitude of the block's acceleration, we need to analyze the forces acting on the block.
1. Determine the force of gravity: The force of gravity acting on the block can be calculated using the formula F_gravity = m * g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2). Given that the mass of the block is 5.2 kg, the force of gravity is 5.2 kg * 9.8 m/s^2 = 50.96 N.
2. Determine the force of friction: The force of friction can be calculated using the formula F_friction = μ * F_N, where μ is the coefficient of friction and F_N is the normal force. Since the block is on the ceiling, the normal force is equal to the force of gravity acting on the block (F_N = F_gravity = 50.96 N). Therefore, the force of friction is F_friction = 0.34 * 50.96 N = 17.34 N.
3. Resolve the applied force: The applied force can be resolved into horizontal and vertical components. The vertical component has no effect on the motion because it is perpendicular to the surface. The horizontal component can be calculated using the formula F_horizontal = F_applied * cos(θ), where θ is the angle between the force and the horizontal. Substituting the given values, F_horizontal = 86 N * cos(61°) = 43.71 N.
4. Calculate the net force: The net force acting on the block can be calculated by subtracting the force of friction from the horizontal component of the applied force. Net force = F_horizontal - F_friction = 43.71 N - 17.34 N = 26.37 N.
5. Calculate the acceleration: The acceleration of the block can be calculated using Newton's second law, F_net = m * a, where F_net is the net force and a is the acceleration. Rearranging the formula, a = F_net / m = 26.37 N / 5.2 kg = 5.07 m/s^2.
Therefore, the magnitude of the block's acceleration is 5.07 m/s^2.