Guided Study: Energy Conservation and Forces on a Curved Ramp

Q1. Throckmorton skateboards from rest down a curved, frictionless ramp. He moves through a quarter-circle with radius $R = 3.00$ m. Throcky and his skateboard have a total mass of $25.0$ kg. (a) Find his speed at the bottom of the ramp. (b) Find the normal force that acts on him at the bottom of the curve.

Background

Topic: Conservation of Energy and Circular Motion Forces

This question tests your understanding of how gravitational potential energy is converted to kinetic energy in a frictionless system, and how to analyze forces at the bottom of a curved path using concepts from circular motion.

Key Terms and Formulas

• Gravitational Potential Energy: $U = mgh$

• Kinetic Energy: $K = \frac{1}{2}mv^2$

• Conservation of Mechanical Energy: $U_{\text{initial}} + K_{\text{initial}} = U_{\text{final}} + K_{\text{final}}$

• For a quarter-circle ramp, the height $h$ is equal to the radius $R$.

• Normal Force at the bottom of a curve: $n = mg + \frac{mv^2}{R}$

Step-by-Step Guidance

1. Start by identifying the initial and final states. Throcky starts from rest at the top (initial kinetic energy is zero), and at the bottom, all potential energy is converted to kinetic energy.

2. Write the conservation of energy equation: $mgh = \frac{1}{2}mv^2$ (since $K_{\text{initial}} = 0$ and $U_{\text{final}} = 0$ at the bottom).

3. Substitute $h = R$ (since the vertical drop is equal to the radius for a quarter-circle ramp): $mgR = \frac{1}{2}mv^2$.

4. Rearrange the equation to solve for $v$: $v = \sqrt{2gR}$.

5. For part (b), recall that at the bottom of the curve, Throcky experiences both the normal force and the centripetal force required for circular motion. The normal force is given by $n = mg + \frac{mv^2}{R}$.

Try solving on your own before revealing the answer!

Final Answer:

(a) $v = 7.67$ m/s

(b) $n = 25.0 \times 9.80 + \frac{25.0 \times (7.67)^2}{3.00}$ N $\approx 408$ N

We used conservation of energy to find the speed, and then applied the formula for normal force at the bottom of a curve, which includes both gravity and the centripetal force.