A 236 kg block is released at a 4.2 m height as shown. The track is frictionless except for a portion of length 6.2 m . The block travels down the track, hits a spring of force constant k = 1483 N/m . The coefficient of kinetic fric- tion between surface and block over the 6.2 m track length is 0.52 .
The acceleration of gravity is 9.8 m/s2 .
To calculate the acceleration of the block as it travels down the track and hits the spring, we will use the following steps:
Step 1: Find the gravitational potential energy of the block at the starting height.
Step 2: Determine the distance over which the block experiences friction.
Step 3: Calculate the work done by the friction force.
Step 4: Determine the net work done on the block.
Step 5: Apply the work-energy principle to find the change in kinetic energy of the block.
Step 6: Use the change in kinetic energy to determine the final velocity of the block.
Step 7: Calculate the acceleration using the final velocity and distance.
Let's go through each step in detail:
Step 1: Find the gravitational potential energy (PE) of the block. The formula for gravitational potential energy is PE = mgh, where m is the mass (236 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (4.2 m). Substitute the given values into the formula:
PE = (236 kg)(9.8 m/s^2)(4.2 m)
PE = 9811.04 Joules
Step 2: Determine the distance over which the block experiences friction. The given information states that the portion of the track with friction is 6.2 m in length.
Step 3: Calculate the work done by the friction force. The formula for work is W = Fd, where F is the force and d is the distance. The friction force is given by the product of the coefficient of kinetic friction (μ) and the normal force (F_norm). The normal force is equal to the weight of the block, which is mg. Therefore, the friction force F_friction = μmg. Substitute the given value for the coefficient of kinetic friction (0.52), the mass (236 kg), and the acceleration due to gravity (9.8 m/s^2) into the formula:
F_friction = (0.52)(236 kg)(9.8 m/s^2)
F_friction = 1188.86 N
The work done by the friction force is therefore:
W_friction = F_friction * d
W_friction = (1188.86 N)(6.2 m)
W_friction = 7364.892 Joules
Step 4: Determine the net work done on the block. The net work done is the sum of the work done by the friction force and the work done by the spring force. Since the track is frictionless except for the portion of length 6.2 m, the only force acting on the block is the spring force. Therefore, the net work done is equal to the work done by the spring force.
Step 5: Apply the work-energy principle to find the change in kinetic energy of the block. The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy. Mathematically, this can be expressed as:
W_net = ΔKE
Since the block starts from rest, its initial kinetic energy is zero (KE_initial = 0). Therefore, the net work done is equal to the final kinetic energy (KE_final). Rearranging the equation, we get:
W_net = KE_final - KE_initial
Since KE_initial = 0, the equation simplifies to:
W_net = KE_final
Substituting the value for the net work done by the spring force (W_net = 7364.892 Joules), we find:
KE_final = 7364.892 Joules
Step 6: Use the change in kinetic energy to determine the final velocity of the block. The formula for kinetic energy is KE = 0.5mv^2, where m is the mass and v is the velocity. Rearranging the equation, we get:
v = √(2 * KE / m)
Substituting the known values for KE_final (7364.892 Joules) and m (236 kg), we can find the final velocity:
v = √(2 * 7364.892 Joules / 236 kg)
v ≈ 8.609 m/s
Step 7: Calculate the acceleration using the final velocity and distance. The formula for acceleration is a = (v_final^2 - v_initial^2) / (2 * d), where v_final is the final velocity, v_initial is the initial velocity (0 m/s since the block starts from rest), and d is the distance. Substituting the known values, we can find the acceleration:
a = (8.609^2 m/s - 0^2 m/s) / (2 * 6.2 m)
a ≈ 0.724 m/s^2
Therefore, the acceleration of the 236 kg block as it travels down the track and hits the spring is approximately 0.724 m/s^2.