A $45.0$-kg crate of tools rests on a horizontal floor. You exert a gradually increasing horizontal push on it, and the crate just begins to move when your force exceeds $313$ N. Then you must reduce your push to $208$ N to keep it moving at a steady $25.0$ cm/s. Suppose you were performing the same experiment on the moon, where the acceleration due to gravity is $1.62$ m/s².
(i) What magnitude push would cause it to move?
(ii) What would its acceleration be if you maintained the push in part (b)? Note: Part (b) asked what push you must exert to give it an acceleration of $1.10$ m/s².

Verified step by step guidance

1. Step 1: Begin by understanding the forces acting on the crate. On Earth, the force required to overcome static friction is given as 313 N. The force required to maintain steady motion is 208 N, which corresponds to the kinetic friction force. The coefficient of static friction (μ_s) and kinetic friction (μ_k) can be calculated using the normal force, which is equal to the weight of the crate (mg). Use the formulas: F_s = μ_s * N and F_k = μ_k * N, where N = mg.
2. Step 2: Calculate the normal force on the moon. The normal force is equal to the weight of the crate, which depends on the moon's gravitational acceleration (1.62 m/s²). Use the formula: N = m * g_moon, where m = 45.0 kg and g_moon = 1.62 m/s².
3. Step 3: Determine the force required to overcome static friction on the moon. Use the coefficient of static friction (μ_s) calculated from the Earth scenario and apply it to the moon's normal force. The formula is: F_s_moon = μ_s * N_moon.
4. Step 4: Calculate the acceleration of the crate on the moon when a constant push of 208 N is applied. First, find the net force acting on the crate by subtracting the kinetic friction force (F_k_moon = μ_k * N_moon) from the applied force. Then, use Newton's second law: a = F_net / m, where F_net is the net force and m is the mass of the crate.
5. Step 5: Summarize the results. The magnitude of the push required to move the crate on the moon is determined by the static friction force (F_s_moon). The acceleration of the crate when maintaining the push of 208 N is calculated using the net force and Newton's second law.

Key Concepts

Friction

Friction is the force that opposes the relative motion of two surfaces in contact. It is dependent on the nature of the surfaces and the normal force acting between them. The static friction force must be overcome to initiate motion, while kinetic friction acts on moving objects. The coefficients of static and kinetic friction are crucial for calculating the forces required to move an object.

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 essential for analyzing how forces affect the motion of objects, especially when considering different gravitational environments.

Weight and Normal Force

Weight is the force exerted by gravity on an object, calculated as the product of its mass and the acceleration due to gravity (W = mg). On the Moon, where gravity is weaker, the weight of the crate will be less than on Earth, affecting the normal force, which is the perpendicular force exerted by a surface to support the weight of an object resting on it. This change in weight influences the frictional forces and the amount of push required to initiate motion.