A 1.5-kg block rests on top of a 7.5kg block. The cord and pulley have negligible mass, and there is no significant friction anywhere.
Part a) What force F must be applied to the bottom block so the top block accelerates to the right at 2.9 m/s^2 ?
Part b) What is the tension in the connecting cord
m1 =1.5 kg, m2 = 7.5 kg, a=2.9 m/s²
(a) The horizontal projections of the equations of motion for each block are
m1•a = T,
m2•a = T-F,
F = (m1+m2) •a = (1.5+7.5) •2.9 = 26.1 N,
(b) T= m2•a - F= 7.5•2.9 – 26.1 = 4.35 N.
Part a) Well, this sounds like a classic physics puzzle! To find the force required to accelerate the top block to the right, we can use Newton's second law (F=ma), where F is the force, m is the mass, and a is the acceleration. In this case, the mass of the top block is 1.5 kg and the acceleration is 2.9 m/s^2. So, the force F required would be F = (1.5 kg)(2.9 m/s^2) = 4.35 N.
Part b) Ah, the tension in the connecting cord, an interesting question. Since we have no friction and negligible mass for the cord and pulley, we can assume that the tension in the connecting cord will be the same for both blocks. So, the tension in the cord would also be 4.35 N. It's like they're sharing the load evenly, just like good buddies!
To solve this problem, we'll start by drawing a free-body diagram for each block.
For the 1.5 kg block (top block):
- There is a force of gravity acting downwards (mg), where m is the mass (1.5 kg) and g is the acceleration due to gravity (9.8 m/s^2).
For the 7.5 kg block (bottom block):
- There is a force of gravity acting downwards (mg), where m is the mass (7.5 kg) and g is the acceleration due to gravity (9.8 m/s^2).
- There is a tension force acting upwards (T) from the connecting cord.
Now, let's solve each part of the problem:
Part a) What force F must be applied to the bottom block so the top block accelerates to the right at 2.9 m/s^2?
Since there is no significant friction, the force applied to the bottom block will be equal to the force required to accelerate the top block.
Applying Newton's second law (F = ma), we have:
F = (mass of top block) × (acceleration of top block)
F = (1.5 kg) × (2.9 m/s^2)
F = 4.35 N
Therefore, the force F that must be applied to the bottom block is 4.35 N.
Part b) What is the tension in the connecting cord?
Since the force applied to the bottom block is equal to the tension in the connecting cord, the tension can be calculated using the same value of force F determined in part a.
Therefore, the tension in the connecting cord is also 4.35 N.
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
Part a) To find the force (F) required to accelerate the top block, we need to consider the forces acting on the system. In this case, the only external force acting on the system is the force applied to the bottom block.
Let's break down the solution into steps:
Step 1: Identify the masses and acceleration.
- The mass of the top block (m1) is 1.5 kg.
- The mass of the bottom block (m2) is 7.5 kg.
- The acceleration of the top block (a) is 2.9 m/s^2.
Step 2: Determine the net force acting on the system.
- The net force is determined by the force applied to the bottom block (F) minus the force of gravity acting on both blocks combined.
- The force of gravity can be calculated by multiplying the total mass of the system (m1 + m2) by the acceleration due to gravity (9.8 m/s^2).
- So, the net force can be written as F - (m1 + m2) * g, where g represents the acceleration due to gravity.
Step 3: Apply Newton's second law and solve for F.
- According to Newton's second law, the net force is equal to the product of the mass and acceleration.
- So, we can write F - (m1 + m2) * g = (m1 + m2) * a.
- Substitute the given values into the equation: F - (1.5 + 7.5) * 9.8 = (1.5 + 7.5) * 2.9
- Simplify the equation and solve for F to get the force required.
Part b) To find the tension in the connecting cord, we need to consider the forces acting on the top block.
Step 1: Identify the forces acting on the top block.
- The tension in the cord (T) and the force of gravity (m1 * g) act on the top block.
Step 2: Apply Newton's second law and solve for T.
- According to Newton's second law, the net force is equal to the product of the mass and acceleration.
- In this case, the net force is T - m1 * g.
- So, we can write T - m1 * g = m1 * a.
- Substitute the given values into the equation: T - 1.5 * 9.8 = 1.5 * 2.9.
- Simplify the equation and solve for T to get the tension in the connecting cord.
By following these steps, you can determine the force required to accelerate the top block and the tension in the connecting cord.