A massless spring of constant k =60.9 N/m is fixed on the left side of a level track. A block of mass m = 0.7 kg is pressed against the spring and compresses it a distance d, as in the figure below. The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius R = 1.3 m. The entire track and the loop-the-loop are frictionless, except for the section of track between points A and B. Given that the coefficient of kinetic friction between the block and the track along AB is ìk=0.1, and that the length of AB is 2.3 m, determine the minimum compression d of the spring that enables the block to just make it through the loop-the-loop at point C. (Hint: The force exerted by the track on the block will be zero if the block barely makes it through the loop-the-loop.)
http://www.jiskha.com/display.cgi?id=1351542188
To determine the minimum compression distance (d) of the spring that enables the block to just make it through the loop-the-loop at point C, we need to consider the forces acting on the block at different points of the track.
Let's break down the problem into different sections:
1. Compression of the spring:
When the block is pressed against the spring, it compresses it a distance d. The force exerted by the spring is given by Hooke's Law: F_spring = k * d, where k is the spring constant. In this case, F_spring = 60.9 N/m * d.
2. Traveling on the level track:
After the block is released, it travels on the level track from the starting point to point A. The only force acting on it is the force due to the compression of the spring, F_spring.
3. Track section AB:
In this section, the block experiences an external force due to the coefficient of kinetic friction (μk = 0.1) between the block and the track. The magnitude of this force is given by F_friction = μk * (mass * gravity), where mass is the mass of the block and gravity is the acceleration due to gravity (9.8 m/s^2). In this case, F_friction = 0.1 * (0.7 kg * 9.8 m/s^2).
4. The circular loop-the-loop:
At point C, the block needs to have enough speed to just make it through the loop. In order to determine the minimum compression distance of the spring, we need to consider the force exerted by the track on the block. At the top of the loop-the-loop, the normal force exerted by the track is zero, and the force due to gravity is the only force acting on the block.
Now, let's explain the steps to find the minimum compression distance, d:
1. Set up an equation for the force exerted by the spring, F_spring = 60.9 N/m * d.
2. Calculate the force due to friction, F_friction = 0.1 * (0.7 kg * 9.8 m/s^2).
3. Use this force to find the velocity of the block at point A using the work-energy principle: F_friction * AB = (1/2) * m * v_A^2, where v_A is the velocity at point A.
4. Use the velocity at point A to find the velocity at point C using the conservation of mechanical energy: (1/2) * m * v_A^2 = (1/2) * m * v_C^2 + m * g * 2R, where v_C is the velocity at point C and R is the radius of the loop-the-loop.
5. Express the velocity at point C in terms of the compression distance, d, using the relationship: v_C = sqrt(2 * g * (R + (3 * d))) since the block will lose some of its potential energy as it travels from point A to point C due to the work done against friction.
6. Set the velocity at point C equal to zero: sqrt(2 * g * (R + (3 * d))) = 0 and solve for d.
This equation will give us the minimum compression distance, d, required for the block to just make it through the loop-the-loop at point C.