A $5.00$ -kg crate is suspended from the end of a short vertical rope of negligible mass. An upward force $F (t)$ is applied to the end of the rope, and the height of the crate above its initial position is given by $y (t) =$ ( $2.80$ m/s)t + ( $0.610$ m/s³)t³. What is the magnitude of $F$ when $t equals 4.00$ s?

검증된 단계별 안내

Step 1: Begin by identifying the forces acting on the crate. The forces include the gravitational force acting downward, which is given by F_gravity = m * g, where m is the mass of the crate (5.00 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²), and the applied force F(t) acting upward.

Step 2: To find the net force acting on the crate, calculate the acceleration of the crate at time t = 4.00 s. The height y(t) is given as a function of time: y(t) = (2.80 m/s)t + (0.610 m/s³)t³. Differentiate y(t) with respect to t to find the velocity v(t), and then differentiate v(t) to find the acceleration a(t).

Step 3: Perform the differentiation. First, find v(t) = dy/dt = 2.80 m/s + 3 * (0.610 m/s³)t². Then, find a(t) = dv/dt = 6 * (0.610 m/s³)t. Substitute t = 4.00 s into a(t) to calculate the acceleration at that specific time.

Step 4: Use Newton's second law, F_net = m * a, to find the net force acting on the crate. Here, m is the mass of the crate (5.00 kg) and a is the acceleration calculated in the previous step. The net force is the difference between the applied force F(t) and the gravitational force F_gravity.

Step 5: Solve for the applied force F(t) using the equation F(t) = F_net + F_gravity. Substitute the values of F_net (from Step 4) and F_gravity = m * g into the equation. This will give the magnitude of the applied force F(t) at t = 4.00 s.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the equation F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing the forces acting on the crate, especially when determining the applied force F(t) at a specific time.

Kinematics of Motion

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. The height function y(t) = (2.80 m/s)t + (0.610 m/s³)t³ provides information about the crate's position over time. By differentiating this function, we can find the velocity and acceleration, which are essential for applying Newton's Second Law to find the force F at t = 4.00 s.

Net Force and Weight

The net force acting on an object is the vector sum of all forces acting on it. For the crate, this includes the upward force F(t) and the downward gravitational force, which is the weight (W = mg). At any time, the net force can be calculated by subtracting the weight from the applied force, allowing us to solve for F(t) when the crate's acceleration is known.