A block sliding along a flat horizontal surface at an initial velocity of 2.0 m/s experiences a constant resistive force of 4.0 N that brings it to rest in a distance of 2.0 m. What is the mass of the block?
Answer (2)
Apply the work-energy theorem to relate the work done by the resistive force to the change in kinetic energy.
Calculate the work done by the resistive force: W = − F ⋅ d .
Calculate the change in kinetic energy: Δ K E = − 2 1 m v 0 2 .
Solve for the mass m : m = v 0 2 2 F d = ( 2.0 ) 2 2 × 4.0 × 2.0 = 4.0 kg .
The mass of the block is 4.0 kg .
Explanation
Problem Analysis We are given that a block is sliding on a flat horizontal surface with an initial velocity of v 0 = 2.0 m/s . A constant resistive force of F = 4.0 N acts on the block, bringing it to rest over a distance of d = 2.0 m . We want to find the mass m of the block.
Work-Energy Theorem The work-energy theorem states that the work done on an object is equal to its change in kinetic energy. In this case, the work done by the resistive force is W = − F ⋅ d , where the negative sign indicates that the force opposes the motion. The change in kinetic energy is Δ K E = 2 1 m v f 2 − 2 1 m v 0 2 , where v f is the final velocity and v 0 is the initial velocity.
Kinetic Energy Change Since the block comes to rest, the final velocity is v f = 0 . Therefore, the change in kinetic energy is Δ K E = − 2 1 m v 0 2 .
Solving for Mass Equating the work done by the resistive force to the change in kinetic energy, we have: − F ⋅ d = − 2 1 m v 0 2 Solving for the mass m , we get: m = v 0 2 2 F d
Calculating the Mass Substituting the given values, we have: m = ( 2.0 m/s ) 2 2 × 4.0 N × 2.0 m = 4.0 m 2 / s 2 16.0 N m = 4.0 kg Therefore, the mass of the block is 4.0 kg .
Final Answer The mass of the block is 4.0 kg .
Examples
Imagine a hockey puck sliding across the ice. If we know the initial speed of the puck, the distance it travels before stopping, and the constant friction force acting against it, we can determine the puck's mass using the same principles. This calculation helps in designing sports equipment or predicting the motion of objects under friction.
The mass of the block is calculated to be 4.0 kg. This value is obtained using the work-energy principle that relates work done by a force to the change in kinetic energy. By calculating the work done against the resistive force and equating it to the change in kinetic energy, we find the mass accordingly.
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