Problem 54 - University Physics with Modern Physics

Question

You are designing a delivery ramp for crates containing exercise equipment. The \(1470-\mathrm{N}\) crates will move at 1.8 \(\mathrm{m} / \mathrm{s}\) at the top of a ramp that slopes downward at \(22.0^{\circ} .\) The ramp exerts a \(550-\mathrm{N}\) kinetic friction force on each crate, and the maximum static friction force also has this value. Each crate will compress a spring at the bottom of the ramp and will come to rest after traveling a total distance of 8.0 \(\mathrm{m}\) along the ramp. Once stopped, a crate must not rebound back up the ramp. Calculate the force constant of the spring that will be needed in order to meet the design criteria.

Step-by-Step Solution

Answer

The spring constant needed is approximately 1650 N/m.

Step 1: Understanding the Problem

First, we need to clarify what is happening as the crate moves down the ramp. The crate, starting with an initial velocity of 1.8 m/s, will slide down a ramp and compress a spring. The spring must stop the crate without it rebounding. The forces acting on the crate include gravity, kinetic friction, and the spring force.

Step 2: Energy Considerations

We use the principle of energy conservation. The initial kinetic energy plus potential energy (due to gravity) will be converted into the work done against friction and the spring potential energy. The initial kinetic energy is given by \[ KE_i = \frac{1}{2}mv^2 \] The potential energy due to gravity is \[ PE_g = mgh \] The work done against friction over the distance \(d\) is \[ W_f = F_{friction} \times d \] And the potential energy stored in the spring when compressed is \[ PE_s = \frac{1}{2}kx^2 \] where \(k\) is the spring constant and \(x\) is the compression.

Step 3: Initial Energies

Calculate the initial kinetic energy using the mass derived from the weight:\[ m = \frac{1470 \text{ N}}{9.8 \text{ m/s}^2} \]Substitute into the kinetic energy formula:\[ KE_i = \frac{1}{2} \left( \frac{1470}{9.8} \right) (1.8)^2 \]

Step 4: Potential Energy and Work Done by Friction

Determine the height \(h\) related to the ramp's slope:\[ h = 8.0 \times \sin(22.0^{\circ}) \]Calculate potenti... and work done by the friction force:\[ PE_g = \frac{1470}{9.8} \times 9.8 \times h = 1470\cdot h \]\[ W_f = 550 \times 8.0 \]

Step 5: Balance Energy Equation

Set up the energy conservation equation:\[ \frac{1}{2}mv^2 + mhg = 550 \times 8 + \frac{1}{2}kx^2 \]This simplifies to:\[ \frac{1}{2} \left(\frac{1470}{9.8}\right)(1.8)^2 +1470\cdot 8 \cdot \sin(22) = 550\cdot 8 + \frac{1}{2}k8^2 \]

Step 6: Solve for the Spring Constant \(k\)

Rearrange the equation to solve for \(k\):\[ \frac{1}{2}k\cdot8^2 = \left( \frac{1}{2} \left( \frac{1470}{9.8} \right)(1.8)^2 + 1470\cdot 8 \cdot \sin(22) - 550 \times 8 \right) \]\[ k = \frac{2\left( KE_i + PE_g - W_f \right)}{8^2} \]Calculate \(k\) by substituting all known values.

Key Concepts

Kinetic Energy

Understanding kinetic energy is crucial in solving physics problems like the one involving the delivery ramp. Kinetic energy (\( KE \)) refers to the energy that an object possesses due to its motion. When a crate slides down the ramp, it converts potential energy into kinetic energy.The kinetic energy is calculated with the formula:\[ KE = \frac{1}{2} mv^2\]- \( m \) is the mass of the object and can be derived from the weight using the equation \( m = \frac{F_g}{g} \)- \( v \) is the velocity of the object, which in this case is the speed of the crate at the top of the ramp.In our problem, the crate starts with an initial velocity of 1.8 m/s. The initial kinetic energy is crucial because as the crate moves down the ramp, this kinetic energy is transferred into other forms of energy, eventually being used to compress the spring at the bottom.

Potential Energy

Potential energy is another vital concept when dealing with objects on an inclined plane. Potential energy (\( PE \)) is the stored energy of an object due to its position relative to a reference point, often the lowest point possible.For an object on a ramp, gravitational potential energy is given by:\[ PE = mgh\]- \( m \) is the mass,- \( g \) is the acceleration due to gravity (\( 9.8 \text{ m/s}^2 \)), and- \( h \) is the height of the ramp.The height can be determined by the angle of the ramp, as shown in the original exercise:\[ h = d \sin(\theta)\]Where \( d \) is the total distance along the ramp and \( \theta \) is the angle of inclination. This potential energy is converted into kinetic energy as the crate moves down, and then into the energy needed to compress the spring.

Spring Force

Spring force is a fundamental concept for understanding how and why the crate comes to rest when it compresses the spring at the bottom of the ramp. The force exerted by a spring is described by Hooke's Law, which says that the force needed to compress or extend a spring is proportional to the distance it is compressed or extended.The formula for spring force is:\[ F_s = kx\]- \( k \) is the spring constant, which measures the stiffness of the spring.- \( x \) is the compression distance.In this exercise, the spring's job is to absorb the kinetic energy of the crate and prevent it from rebounding. The energy stored in the spring as potential energy is given by:\[ PE_s = \frac{1}{2} kx^2\]By setting the potential energy in the spring equal to the net energy after accounting for friction, we can solve for \( k \). This calculation ensures that not only does the crate stop, but it also does not bounce back, achieving the desired design criteria.