Practice Questions

A child pulls a toy car of mass $0.5$ kg across a rough horizontal floor with a force of $2$ N directed at an angle of $37^\circ$ above the horizontal. If the car moves a distance of $5$ m, calculate the work done by the child. (Given $\cos 37^\circ = 0.8$)

A force $\mathbf{F} = (2\hat{\mathbf{i}} + 5\hat{\mathbf{j}} - 3\hat{\mathbf{k}})$ N acts on a particle, causing a displacement $\mathbf{d} = (4\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 6\hat{\mathbf{k}})$ m. Calculate the work done by the force.

Define work done by a constant force.

A spring with a spring constant $k = 500 \text{ N/m}$ is compressed by $10$ cm. Calculate the potential energy stored in the spring.

Recall the formula for kinetic energy and name its SI unit.

Propose a scenario where a net force acts on an object, but the work done by this force is zero over a certain displacement.

Evaluate the work done by the gravitational force on a satellite in a perfect circular orbit around the Earth over one complete revolution. Justify your answer using the definition of work as a scalar product.

Define power and state its SI unit.

Propose a modification to a standard one-dimensional collision experiment (e.g., using air track gliders) to investigate the difference between elastic and completely inelastic collisions.

Define the scalar product (or dot product) of two vectors $\mathbf{A}$ and $\mathbf{B}$. Write the formula and explain what each term represents.

A machine gun fires $360$ bullets per minute, each with a speed of $600 \text{ m/s}$. If the mass of each bullet is $50$ g, calculate the power of the gun.

A car of mass $1500 \text{ kg}$ is moving at a speed of $18 \text{ km/h}$. Recall the formula for kinetic energy and calculate its value.

A particle moves from a point $\mathbf{r}_i = (2\hat{\mathbf{i}} + 3\hat{\mathbf{j}})$ m to a point $\mathbf{r}_f = (3\hat{\mathbf{i}} + \hat{\mathbf{j}})$ m under the action of a constant force $\mathbf{F} = (5\hat{\mathbf{i}} + 5\hat{\mathbf{j}})$ N. Calculate the work done.

State and explain the work-energy theorem.

Explain the difference between an elastic collision and a completely inelastic collision. Give one example for each.

List three different units used to measure work or energy.

A body of mass $m$ has kinetic energy $K$. What is its linear momentum in terms of $K$ and $m$? Apply this to compare the momenta of a proton and an alpha particle if they have the same kinetic energy.

A raindrop of mass $1.5$ g falls from a height of $500$ m and hits the ground with a speed of $40 \text{ m/s}$. Apply the work-energy theorem to calculate the work done by the resistive force of the air. (Take $g = 10 \text{ m/s}^2$)

A student claims that in an elastic collision between two unequal masses, it is impossible for the lighter mass to remain at rest after the collision if it was initially moving and collided with a stationary heavier mass. Evaluate this claim and justify your conclusion.

A $1$ kg block moving at $4 \text{ m/s}$ on a frictionless surface collides head-on with a stationary $3$ kg block. If the collision is elastic, calculate the velocities of both blocks after the collision.

Examine the statement: 'In an inelastic collision, momentum is conserved but kinetic energy is not.' Is total energy conserved? Justify your answer.

A light body and a heavy body have the same momentum. Compare their kinetic energies.

Design an experiment to verify the principle of conservation of mechanical energy using a simple pendulum. Justify your choice of measurements and formulate the expected results.

Justify why the concept of potential energy is not defined for a non-conservative force like friction.

Design a system involving a spring and a block on an inclined plane to demonstrate the conversion between gravitational potential energy, elastic potential energy, and kinetic energy. Justify how you would show that total mechanical energy is conserved if the plane is frictionless.

A student argues that in a ballistic pendulum experiment (a bullet fired into a stationary block), the initial kinetic energy of the bullet is equal to the final maximum potential energy of the block-bullet system. Critique this argument.

Evaluate the factors that determine the minimum power required for a motor to lift an elevator. Formulate an equation for the instantaneous power delivered by the motor if the elevator accelerates upwards with an acceleration $a$, considering a frictional force $f$.

Design an experiment to determine the spring constant $k$ of a spring using the principle of conservation of energy. Justify why this method might be more accurate than using Hooke's Law statically.

A box of mass $20 \text{ kg}$ is lifted from the ground to a height of $2.5 \text{ m}$. Taking the acceleration due to gravity $g = 9.8 \text{ m/s}^2$, calculate the potential energy stored in the box.

State Hooke's law for an ideal spring. Explain how potential energy is stored in a spring and provide the mathematical expression for it.

Explain the concept of potential energy. Provide the formula for gravitational potential energy near the Earth's surface.

List two key properties of a conservative force.

A motor pulls an object along a horizontal surface at a constant velocity of $3 \text{ m/s}$ by applying a constant horizontal force of $500 \text{ N}$. Recall the formula for power and calculate the power delivered by the motor.

A block of mass $2$ kg is pushed up an inclined plane of inclination $30^\circ$ over a distance of $10$ m. The force applied parallel to the incline is $15$ N. If the coefficient of kinetic friction is $0.1$, calculate the work done by the applied force, the gravitational force, and the frictional force. (Assume $g = 9.8 \text{ m/s}^2$)

A car of mass $1200$ kg moving at $15 \text{ m/s}$ collides with a stationary car of mass $800$ kg. After the collision, they stick together. Calculate their common velocity and the loss in kinetic energy during the collision.

Analyze why the work done by a centripetal force in maintaining uniform circular motion is always zero.

A particle is moving in a potential field described by the function $V(x) = \frac{A}{x^2} - \frac{B}{x}$, where $A$ and $B$ are positive constants. Create a qualitative plot of this potential energy. Formulate the expression for the force $F(x)$ acting on the particle. Evaluate the position $x_0$ where the particle is in stable equilibrium.

Formulate a potential energy function $V(x)$ for a hypothetical conservative force given by $F(x) = -kx + cx^3$, where $k$ and $c$ are positive constants. Assuming the potential energy is zero at $x=0$, create a qualitative plot of $V(x)$ versus $x$, identifying the equilibrium points.

Critique the statement: "The work-energy theorem is not a fundamental principle but just a mathematical consequence of Newton's second law, offering no new physical insight."

A bystander proposes that in an inelastic collision between a moving truck and a stationary car, the change in kinetic energy is the same for both vehicles because the forces they exert on each other are equal and opposite. Evaluate this proposal and critique the bystander's reasoning.

A pump is required to lift $600$ kg of water per minute from a well $25$ m deep and eject it with a speed of $50 \text{ m/s}$. Calculate the power required to perform this task. (Assume $g = 9.8 \text{ m/s}^2$)

A constant force $\mathbf{F} = (2\hat{\mathbf{i}} + 3\hat{\mathbf{j}} - \hat{\mathbf{k}}) \text{ N}$ acts on a body, producing a displacement of $\mathbf{d} = (4\hat{\mathbf{i}} + \hat{\mathbf{j}} + 3\hat{\mathbf{k}}) \text{ m}$. Recall the formula for work done and calculate the work done by this force.

Describe the three conditions under which the work done by a force on an object is zero. Provide a simple physical example for each condition.

The potential energy of a particle in a certain field is given by $V(x) = \frac{A}{x^2} - \frac{B}{x}$, where $A$ and $B$ are positive constants. Calculate the force $F(x)$ acting on the particle.

Formulate a real-world example where positive work is done by a kinetic frictional force.